Toric Varieties David Cox John Little Hal Schenck D EPARTMENT 01002
OF
M ATHEMATICS , A MHERST C OLLEGE , A MHERST, MA
E-mail address:
[email protected] D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE , C OLLEGE THE H OLY C ROSS , W ORCESTER , MA 01610
OF
E-mail address:
[email protected] D EPARTMENT OF M ATHEMATICS , U NIVERSITY C HAMPAIGN , U RBANA , IL 61801 E-mail address:
[email protected] OF I LLINOIS AT
U RBANA -
c 2009, David Cox, John Little and Hal Schenck
Preface
The study of toric varieties is a wonderful part of algebraic geometry that has deep connections with polyhedral geometry. Our book is an introduction to this rich subject that assumes only a modest knowledge of algebraic geometry. There are elegant theorems, unexpected applications, and, as noted by Fulton [30], “toric varieties have provided a remarkably fertile testing ground for general theories.” The Current Version. The January 2009 version consists of seven chapters: Chapter 1: Affine Toric Varieties Chapter 2: Projective Toric Varieties Chapter 3: Normal Toric Varieties Chapter 4: Divisors on Toric Varieties Chapter 5: Homogeneous Coordinates Chapter 6: Line Bundles on Toric Varieties Chapter 7: Projective Toric Morphisms These are the chapters included in the version you downloaded. The book also has a list of notation, a bibliography, and an index, all of which will appear in more polished form in the published version of the book. Two versions are available on-line. We recommend using postscript version since it has superior quality. Changes to the August 2008 Version. The new version fixes some typographical errors and has improved running heads. Other additions include: Chapter 4 includes more sheaf theory. Chapter 4 has an exercise about support functions and tropical polynomials. Chapter 5 now discusses sheaves associated to a graded modules. iii
Preface
iv
The Rest of the Book. Five chapters are in various stages of completion: Chapter 8: The Canonical Divisor of a Toric Variety Chapter 9: Sheaf Cohomology of Toric Varieties Chapter 10: Toric Surfaces Chapter 11: Toric Singularities Chapter 12: The Topology of Toric Varieties When the book is completed in August 2010, there will be three final chapters: Chapter 13: The Riemann-Roch Theorem Chapter 14: Geometric Invariant Theory Chapter 15: The Toric Minimal Model Program Prerequisites. The text assumes the material covered in basic graduate courses in algebra, topology, and complex analysis. In addition, we assume that the reader has had some previous experience with algebraic geometry, at the level of any of the following texts: Ideals, Varieties and Algorithms by Cox, Little and O’Shea [17] Introduction to Algebraic Geometry by Hassett [43] Elementary Algebraic Geometry by Hulek [52] Undergraduate Algebraic Geometry by Reid [84] Computational Algebraic Geometry by Schenck [90] An Invitation to Algebraic Geometry by Smith, Kahanp¨aa¨ , Kek¨al¨ainen and Traves [94] Readers who have studied more sophisticated texts such as Harris [40], Hartshorne [41] or Shafarevich [89] certainly have the background needed to read our book. We should also mention that Chapter 9 uses some basic facts from algebraic topology. The books by Hatcher [44] and Munkres [72] are useful references. Background Sections. Since we do not assume a complete knowledge of algebraic geometry, Chapters 1–9 each begin with a background section that introduces the definitions and theorems from algebraic geometry that are needed to understand the chapter. The remaining chapters do not have background sections. For some of the chapters, no further background is necessary, while for others, the material more sophisticated and the requisite background will be provided by careful references to the literature. The Structure of the Text. We number theorems, propositions and equations based on the chapter and the section. Thus §3.2 refers to section 2 of Chapter 3, and Theorem 3.2.6 and equation (3.2.6) appear in this section. The end (or absence) of a proof is indicated by , and the end of an example is indicated by .
Preface
v
For the Instructor. We do not yet have a clear idea of how many chapters can be covered in a given course. This will depend on both the length of the course and the level of the students. One reason for posting this preliminary version on the internet is our hope that you will teach from the book and give us feedback about what worked, what didn’t, how much you covered, and how much algebraic geometry your students knew at the beginning of the course. Also let us know if the book works for students who know very little algebraic geometry. We look forward to hearing from you! For the Student. The book assumes that you will be an active reader. This means in particular that you should do tons of exercises—this is the best way to learn about toric varieties. For students with a more modest background in algebraic geometry, reading the book requires a commitment to learn both toric varieties and algebraic geometry. It will be a lot of work, but it’s worth the effort. This is a great subject. What’s Missing. Right now, we do not discuss the history of toric varieties, nor do we give detailed notes about how results in the text relate to the literature. We would be interesting in hearing from readers about whether these items should be included. Please Give Us Feedback. We urge all readers to let us know about: Typographical and mathematical errors. Unclear proofs. Omitted references. Topics not in the book that should be covered. Places where we do not give proper credit. As we said above, we look forward to hearing from you! January 2009
David Cox John Little Hal Schenck
Contents
Preface
iii
Notation
xi
Part I: Basic Theory of Toric Varieties
1
Chapter 1.
3
Affine Toric Varieties
§1.0.
Background: Affine Varieties
§1.1.
Introduction to Affine Toric Varieties
10
§1.2.
Cones and Affine Toric Varieties
22
§1.3.
Properties of Affine Toric Varieties
34
Appendix: Tensor Products of Coordinate Rings Chapter 2.
Projective Toric Varieties
3
48 49
§2.0.
Background: Projective Varieties
49
§2.1.
Lattice Points and Projective Toric Varieties
55
§2.2.
Lattice Points and Polytopes
63
§2.3.
Polytopes and Projective Toric Varieties
75
§2.4.
Properties of Projective Toric Varieties
86
Chapter 3.
Normal Toric Varieties
93
§3.0.
Background: Abstract Varieties
93
§3.1.
Fans and Normal Toric Varieties
106
§3.2.
The Orbit-Cone Correspondence
115
§3.3.
Equivariant Maps of Toric Varieties
124
§3.4.
Complete and Proper
138 vii
Contents
viii
Appendix: Nonnormal Toric Varieties Chapter 4.
Divisors on Toric Varieties
148 153
§4.0.
Background: Valuations, Divisors and Sheaves
153
§4.1.
Weil Divisors on Toric Varieties
169
§4.2.
Cartier Divisors on Toric Varieties
174
§4.3.
The Sheaf of a Torus-Invariant Divisor
187
Chapter 5.
Homogeneous Coordinates
193
§5.0.
Background: Quotients in Algebraic Geometry
193
§5.1.
Quotient Constructions of Toric Varieties
202
§5.2.
The Total Coordinate Ring
216
§5.3.
Sheaves on Toric Varieties
224
§5.4.
Homogenization and Polytopes
229
Chapter 6.
Line Bundles on Toric Varieties
243
§6.0.
Background: Sheaves and Line Bundles
243
§6.1.
Ample Divisors on Complete Toric Varieties
260
§6.2.
The Nef and Mori Cones
279
§6.3.
The Simplicial Case
289
Appendix: Quasicoherent Sheaves on Toric Varieties Chapter 7.
Projective Toric Morphisms
299 303
§7.0.
Background: Quasiprojective Varieties and Projective Morphisms
303
§7.1.
Polyhedra and Toric Varieties
307
§7.2.
Projective Morphisms and Toric Varieties
315
§7.3.
Projective Bundles and Toric Varieties
321
Appendix: More on Projective Morphisms Chapter 8.
The Canonical Divisor of a Toric Variety
332 337
§8.0.
Background: Reflexive Sheaves and Differential Forms
337
§8.1.
One-Forms on Toric Varieties
347
§8.2.
p-Forms on Toric Varieties
352
§8.3.
Fano Toric Varieties
352
Chapter 9.
Sheaf Cohomology of Toric Varieties
353
§9.0.
Background: Cohomology
353
§9.1.
Cohomology of Toric Line Bundles
365
§9.2.
Serre Duality
380
Contents
ix
§9.3.
The Bott-Steenbrink-Danilov Vanishing Theorem
385
§9.4.
Local Cohomology and the Total Coordinate Ring
385
Appendix: Introduction to Spectral Sequences
395
Topics in Toric Geometry
401
Chapter 10.
403
Toric Surfaces
§10.1.
Singularities of Toric Surfaces and Their Resolutions
403
§10.2.
Continued Fractions and Toric Surfaces
412
§10.3.
Gr¨obner Fans and McKay Correspondences
423
§10.4.
Smooth Toric Surfaces
433
§10.5.
Riemann-Roch and Lattice Polygons
441
Chapter 11.
Toric Singularities
453
§11.1.
Existence of Resolutions
453
§11.2.
Projective Resolutions
461
§11.3.
Blowing Up an Ideal Sheaf
461
§11.4.
Some Important Toric Singularities
461
Chapter 12.
The Topology of Toric Varieties
463
Chapter 13.
The Riemann-Roch Theorem
465
Chapter 14.
Geometric Invariant Theory
467
Chapter 15.
The Toric Minimal Model Program
469
Bibliography
471
Index
477
Notation
Basic Notions
! "!# $ "
integers, rational numbers, real numbers, complex numbers semigroup of nonnegative integers image and kernel direct limit inverse limit
Rings and Varieties
&% ')(*',+.&%0% ' ( *' + -0&% '21( ( *' +1 ( 3547698 :.4;&%>;-? &4; -
F%>;(*-
A%>;(
/
; -
%
&%>; - A% -
.
F% -
' with -algebra homomorphisms 'M , there should be a unique M algebra homomorphism (the dotted arrow) that makes the diagram commute. By A%>; ( - 1 F%>; the universal mapping property of the tensor product of -algebras, F > % ; * ( =&%>; has the mapping properties we want. Since is a finitely generated -algebra with no nilpotents (see the appendix to this chapter), it is the coordinate F > % ; (
; ring . For more on tensor products, see [2, pp. 24–27] or [25, A2.2].
;
$
Example 1.0.10. Let be an affine variety. Since ; + has product coordinate ring
+ R BC D.47A% ( + - 8 , the
A%>;S- &% ( + - R A%>;S- % ( + -;-8R+ A % ; -; - . If . 5 , then the action be the subspace spanned by the . Thus O . 9 F%>; - , so that we get O . A%>;S- . ; of on induces an action on the coordinate ring
§1.1. Introduction to Affine Toric Varieties
19
5
By Definition 1.1.3, this extends the usual P 4 8 of' P on , which in terms of the ' F% - is given by ' P R action coordinate ring ; that and hence 9 F%>; - are stable under the action of 5 . Since .9 ItAfollows %>;S- is finite-dimensional, A%>;S- is spanned by simultaneous eigenvectors 9 Proposition 1.1.2 implies that 5 . But this is taking place in F% ; - , where the simultaneous eigenvectors are of 9 A%>;S- is ' spanned characters. Then the above the characters! It follows that P &%>by A > % S ; . ;- for , . . It follows that expression for O . implies that 9 O . F% 2 - , proving that A%>;S- R A% 2 - .
A%>;S-
2
It remains to show that is finitely generated. Since is finitely generated, ( O " . &%>;- with &%>;- R &% O ( O " - . Expressing ' we can find O 2 each O 2 in terms of characters as above gives the desired finite generating set of . Hence is an affine semigroup. Here is one way to think about the above proof. When an irreducible affine ; 5 as a Zariski open subset, we have the inclusion variety contains a torus
A%>;S- ! F % ; -; -
5
Thus consists of those functions on the torus that extend to polynomial ; ; functions on . Then the key insight is that is a toric variety precisely when the functions that extend are determined by the characters that extend.
;
8
354 ' R
R ! Example 1.1.17. We’ve seen that variety ' R A % ' X- . Also, the torus 47 isM 8 a toric L ( is via the map with toric ideal K ! 4 ( 8 4 ( ( " 8 . The lattice points used in this map can be represented as the columns of the matrix
(1.1.6)
2
R
! The corresponding semigroup consists of the -linear combinations of 2 the column vectors. Hence the elements of are lattice points lying in the poly pictured in Figure 1 on the next page. In this figure, the four hedral region in 2 vectors generating are shown in bold, and the boundary of the polyhedral region is partially shaded. In the terminology of §1.2, this polyhedral region is a rational 2 polyhedral cone. In Exercise 1.1.5 you will show that consists of all lattice points ; lying in the cone in Figure 1. We will use this in §1.3 to prove that is normal.
Exercises for §1.1.
$( d + d
# d + d Om #f -IH d 27+6+7+62 d J
1.1.1. As in Example 1.1.6, let
and let
g be the surface parametrized by F12nn'B(MF 23 26+7+6+92 6 2 'a +
Chapter 1. Affine Toric Varieties
20
(0,0,1)
(0,1,0)
(1,0,0)
(1,1,−1)
Figure 1. Cone containing the lattice points corresponding to
'Z( F 4 'a . Thus g ( ' . (b) Prove that g ' is homogeneous.
(c) Consider lex monomial order with d *d *d . Let ,% g ' be homogeneous of degree and let be the remainder of , on division by the generators of . Prove that can be written ( d 2 d ' . d 2 d 4 ' . d 2 d ' (a) Prove that
d # 27+6+7+6237 2n ')(C to show that I(V . (d) Use part (c) and WF 23 (e) Use parts (b), (c) and (d) to prove that $ ( g ' . Also explain why the generators of are a Gr¨obner basis for the above lex order. 1.1.2. Let - Prove that d"!$# be ad sublattice. & % # O U' ( d r # d)( OsB+2 * , ! 2 s-# * . + Note that when y. , the vectors 2 . , ! have disjoint! support (i.e., no coordinate is , with s-# *%' . positive in both), while this may fail for arbitrary sZ/2 * ' 1.1.3. Let 0 be a toric ideal and let 26+7+6+92 1 be a basis of the sublattice "! . Define 32 ( d 54# # d $4% O (L126+6+7+9/2 + Prove + 0 ( 3276 d 8pd ! . Hint: Given sZ/2 * . , ! with s-# *%. , write s# *( 9 +1 that + , + . This implies > < > < d r ( #f(;: dd $44#%? 4 : dd $ 44# % ? 4 #fm+ < 45= < 4$@ d d Show that multiplying this by ! +' A gives an element of 82 for BC . (By being where + is homogeneous of degree . Also explain why we may assume that the coefficient of + in ,+ is zero for .
more careful, one can show that this result holds for lattice ideals. See [68, Lem. 7.6].)
§1.1. Introduction to Affine Toric Varieties
21
G , 6 2 7 + 6 + 9 + 2 , f I H : G J . This gives a polynomial map 6 ! GMh ! 61H d 27+6+6+72 d ! Je#h H G:J"2 d + j#h , + + Let f ! be the Zariski closure of the image of . (a) Prove that 'B(:?7 '. (b) Explain how this applies to the proof of Proposition 1.1.14. 1.1.5. Let ( p12 `23 1'92 4 ( A 2612 1'92 o (LA 2 26E'92 (Mnm26126#&' be the columns 1.1.4. Fix an affine variety and , which on coordinate rings is given by
of the matrix in Example 1.1.17 and let
g (
+ + O +
+
o
g Uo is a semigroup generated by 2 4 2 o 2 .
be the cone in Figure 1. Prove that
1.1.6. An interesting observation is that different sets of lattice points can parametrize the same affine toric variety, even though these parametrizations behave slightly differently. In this exercise you will consider the parametrizations
4 o and F 1 n 2 n Z ' L ( F 3 2 7 3 2 7 ' 4 F12nn'B(M F o 237 2nd o ' + c o (a) Prove that and both give the affine toric variety ( % # o7'^- . 4 (b) We can regard and as maps 4 61 4 #h and 4 6m 4 #h + Prove that is surjective and that is not. 4 6W ! " . The image of In general, a finite subset gives a rational map # in ! is called a toric set in the literature. Thus A 4 ' and A 4 ' are toric sets. The 4 papers [57] and [85] study when a toric set equals the corresponding affine toric variety.
1.1.7. In Example 1.1.6 and Exercise 1.1.1 we constructed the rational normal cone g using all monomials of degree in 12n . If we drop some of the monomials, things become more complicated. For example, consider the surface parametrized by 2 o 237 o 2n ' + F 1 n 2 n ) ' ( F This gives a toric variety - . Show that the toric ideal of is given by ')( %d $ # &c 2 'c $ 4 # o 2 "d 4 # c 4 $ 2 d 4 # c o fIH d 2 c 2 2 $ J + The toric ideal for g has quadratic generators; by removing the monomial 4 4 , we now
get cubic generators. In Chapter 2 we will use this example to construct a projective curve that is normal but not projectively normal.
B G B F12n 2 S')(MF 2n 2 23,+9P2n 4 o 'a'B ) + (a) Find generators for the toric ideal ( G$'^ BQH d 2 d 2 d 2 d 2 d J . 4 o ) (b) Show that zW{}|-GM(.- . You may assume that Proposition 1.1.8 holds over B . (c) Show that $(0/ d d + d 2 d d 4 d o . o ) 4 o
1.1.8. Instead of working over , we will work over an algebraically closed field *) parametrized by characteristic ( . Consider the affine toric variety
of
Chapter 1. Affine Toric Varieties
22
G0 B
G
) has codimension two and can be defined by two equations, i.e., It follows that is a set-theoretic complete intersection. The paper [3] shows that if we replace with an algebrically closed field of characteristic ( , then the above parametrization is never a set-theoretic complete intersection.
B
K
80
f !
!
1.1.9. Prove that a lattice ideal for is a toric ideal if and only if is torsionfree. Hint: When is torsion-free, it can be regarded as the character lattice of a torus. The other direction of the proof is more challenging. If you get stuck, see [68, Thm. 7.4].
!
$( d 4 #fm2 dc # 12 &c # is the lattice ideal for the lattice t(N/> 2 2 'Z' o O 0 :|>z (W8: o +
1.1.10. Prove that
Also compute primary decomposition of to show that is not prime.
~
*~ gives an ] '( !n' ] is a group )jh ] ~ 1|S 23 ' + 1.1.12. Consider tori ~ and ~ with character lattices and . By Example 1.1.13, 4 H J and IH J . Let 6m~ 4 h ~ be a morphism the coordinate rings of ~ and ~ are I 4 4 4 that is a group homomorphism. Then induces maps
6 4 #h and 6mIH 4 Je#Sh IH J
1.1.11. Let be a torus with character lattice . Then every point
. Prove that evaluation map defined by induces a group isomorphism homomorphism and that the map
] 6 h
by composition. Prove that affine semigroups.
is the map of semigroup algebras induced by the map
of
N( ( %( - , :/E W8 t
1.1.13. A commutative semigroup is cancellative if # $ implies - $ for $ all and torsion-free if implies for all and . Prove that is affine if and only if it is finitely generated, cancellative, and torsion-free.
P2 2
Q%(
1.1.14. The requirement that an affine semigroup be finitely generated is important since lattices contain semigroups that are not finitely generated. For example, let be irrational and consider the semigroup
(N/> 2 9'Z', 4 O *8$ 4 +
Prove that is not finitely generated. (When satisfies a quadratic equation with integer coefficients, the generators of are related to continued fractions. For example, when and ( is the golden ratio, the minimal generators of are for , where is the th Fibonacci number. See [95] for further details.) (
(Mn '
nm23 1' 4 2 4 ' (Mm2 `26+7+6+ 1.1.15. Suppose that ] 6, h is a group isomorphism. Fix a finite set and and #" are equivariantly isomorphic let ! ( ]P ' . Prove that the toric varieties (meaning that the isomorphism respects the torus action).
§1.2. Cones and Affine Toric Varieties We begin with a brief discussion of rational polyhedral cones and then explain how they relate to affine toric varieties.
§1.2. Cones and Affine Toric Varieties
23
Convex Polyhedral Cones. Fix a pair of dual vector spaces ; @ and ? @ . Our discussion of cones will omit most proofs—we refer the reader to [30] for more details and [76, App. A.1] for careful statements. See also [10, 38, 87].
@
Definition 1.2.1. A convex polyhedral cone in ?
= 6 R I> 4 4.8 R
.
One easily checks that a convex polyhedral cone 6 is in fact convex, meaning ' . 6 ) ' 4 1R)28 . 6 for all ) , and is a cone, meaning ' . 6 ) ' . 6 for all ) W . Since we will only consider convex cones, the cones satisfying Definition 1.2.1 will be called simply “polyhedral cones.”
that
of polyhedral cones include the first quadrant in = I> 4 ( ( 8 or first octant in . Examples For another example, the cone is pictured ! in Figure 2 below. It is also possible to have cones that contain entire lines. For = I> 4 (4R ( 8 = I> 4 (4R ( 8 ' 4 ' 8 ! is the -axis, is the closed example, while W . upper half-plane
. As we will see below, these last two examples are not strongly convex. We can also create cones using polytopes, which are defined as follows. Definition 1.2.2. A polytope in ?@ is a set of the form
where
L ! ? @
R = I>)4)4 4 4 ( (
8
Example 1.2.19. Let 6 R ! ?@ R with ? R . This is the cone pictured in Figure 2. By Example 1.2.9, 6 7 is the cone pictured in Figure 1, and by Example 1.1.17, the lattice points in this cone are generated 4 3 by columns of matrix (1.1.6). It follows from Example 1.1.17 that is the affine 5 3 4 ' R 8 . toric variety
+ / and set 6 R = >I 4 ( 8 ! = 6 7 R >I 4 ( U ( + 8
Here are two further examples of Theorem 1.2.18. Example 1.2.20. Fix
3 R BC D.47A% ')( *' * ' 1 U ( ( *' +1 ( - 8 R 5 47 M 8 + " + (Exercise 1.2.7). This implies the general fact that if 6 ! ? @ + 4 3 547 M 8 " . cone of dimension , then and the corresponding affine toric variety is 4
. Then
is a smooth
R and / R . = >I 4 % ( R 8 ! . @ Example 1.2.21. Fix a positive integer @ and let 6 R = I> 4 ( ( 8 This has dual cone 6 7 R 7 on the next page shows 6 7 L 3 @ . Figure R R when @ . The affine semigroup 6 7 9 is generated by the lattice points Figure 5 illustrates the cones in Example 1.2.20 when
Chapter 1. Affine Toric Varieties
32
Figure 7. The cone
4 8
when
"
@ . When @ R , these are the white dots in Figure 7. (You will for prove these assertions in Exercise 1.2.8.)
By §1.1, the variety 47 M 8 affine ? U ( toric map defined by
4
3
is the Zariski closure of the image of the
4 8 R 4 . ? 8
4 8 4 ? ? " ( ? " ( ? 8 This map has the same image as the map ?U ( ? 4 3 used ! in Example 1.1.6. Thus is isomorphic to the rational normal cone minors of the matrix whose ideal is generated by the ' ')( '2? " ',? " ( ')( ' '2? " ( ',? Note that the cones
6
and 6
7
are simplicial but not smooth.
We will return to this example often. One thing evident in Example 1.1.6 is the difference between cone generators and semigroup the cone 6 7 has L 3 R 6 7 9 has generators: two generators but the semigroup @ .
L 3
R 67Q9 ; When 6 ! ? @ has maximal dimension, the semigroup unique minimal generating set constructed as follows. Define an element , 23 Z , Z Z for , Z7 , Z Z . 23 implies , Z R or , to be irreducible if , R ,
Proposition 1.2.22. Let 6 23 R 6 7 9-; . Then
;S-NR A% 5 ; - R &% 5 - 1&A%>;S- R F % ; - 1=A% 2 -;S-
R (a) (b): If is normal, then is integrally closed in its field 2 A < 4 & ; 8 S& and , . ; . , . of fractions . Suppose that for some . P' 5 ; Then is a polynomial function on and hence a rational function on since P 5P ! ; is Zariski open. We also have ' . P F% 2 - since , . 2 . It follows that ' R ' F% 2 - . By the is a root of the monic polynomial in P A% 2 - with2 coefficients 2 ' . definition of normal, we obtain , i.e., ,. . Thus is saturated. 2
$
2
2
(b) (c): Let rational = I> ! 4 $ 8 be a finite generating B %$ setR of . Then !lies = inI> the 4 $ 8 R / ! ; @ , and = I> 4 $ 8 / implies polyhedral cone 7 ! ? @ is a strongly convex by Exercise 1.2.6. It follows that 6 R 2 rational polyhedral cone such that ! 6 7 9; . In Exercise 1.3.4 you will prove 2 2 2 3 . that equality holds when is saturated. Hence R
F% 2 3 -
&%
-
R 6 7 9T; is normal when 6J! ?G@ (c) (a): We need to show that ( be the rays of 6 . Since is a strongly convex rational polyhedral cone. Let 6 is generated by its rays (Lemma 1.2.15), we have Intersecting with ;
gives
2 3 R
'7 ( ' (
6 7 R
(2 ')( , which easily implies
F% 23 - R F%>L 3 -
')( (
F% 2 7 - 4 ( 8 , so that ( (
% 2 - R F% ')(*'V1 *' +1 F &% ' ( *' + - is normal (it is a UFD), so its localization by Example 1.2.20. But A% ')( *' +.- R A% ')(*' 1 ( *' +1 ( 354 ' R 8
is also normal by Exercise 1.0.7. This completes the proof.
;
Example 1.3.6. We saw in Example 1.2.19 that R is the affine toric = I> 4 ( . ( 8 4 3 R of the cone 6 variety pictured in Figure 1. Then ; Theorem 1.3.5 implies that is normal, as claimed in Example 1.1.5. ? (
? ! U
Example 1.3.7. By Example 1.2.21, the rational normal cone is the affine toric variety of a strongly convex rational polyhedral cone and hence is normal by Theorem 1.3.5. It is instructive to view this example ? ? using ? ( ? ( the parametrization
4 . 8 R 4 " " 8 3
$
from Example 1.1.6. Plotting the lattice points in for @ R white 2 gives #$ the squares in Figure 9 (a) below. These generate the semigroup R , and the = I> 4 ( 8 , which is the first quadproof of Theorem 1.3.5 gives the cone 6 7 R rant in the figure. At first glance, something seems wrong. The affine variety is normal, yet in Figure 9 (a) the semigroup generated by the white squares misses some lattice points in 6 7 . This semigroup does not look saturated. How can the affine toric variety be normal?
σ
(a)
σ
(b)
Figure 9. Lattice points for the rational normal cone
The problem is that we are using the wrong lattice! Proposition 1.1.8 tells us %$ to use the lattice , which gives the white dots and squares in Figure 9 (b). This figure shows that the white squares generate the semigroup of lattice points in 6 7 . 2 Hence is saturated and everything is fine.
§1.3. Properties of Affine Toric Varieties
39
This example points out the importance of working with the correct lattice. The Normalization of an Affine Toric Variety. The normalization of an affine toric ; BC D47&% 2 - 8 for an affine semigroup 2 , so that variety is easy to describe. Let R 2 = I> 4 2 8 ; the torus of has character lattice ; R . Let denote the cone of any 2 = I> 4 2 8 R 7 !? @ . In Exercise 1.3.6 you will finite generating set of and set 6 prove the following. Proposition 1.3.8. The above cone 6 is a strongly convex rational polyhedral cone F% 2 - ! A% 687A9 ; - induces a morphism 4 3 ; that is in ? @ and the inclusion ; the normalization map of .
3
The normalization of an affine toric variety of the form is constructed by 1$ %$ applying Proposition 1.3.8 to the affine semigroup and the lattice . Example 1.3.9.
. Then considered in Exercise 1.1.7. This is almost
$ R 4 8 4 8 4 8 4 8 ! Let 3 4 8 R 4 8 3 !
parametrizes the surface ( 4 rational the normal cone , except that we have omitted 8 4 8 , we see that 1$ is not saturated, so that
3
4 .8 R
. Using is not normal.
3
is . This Applying Proposition 1.3.8, one sees that the normalization of , and the normalization map is induced by the is an affine variety in obvious projection .
In Chapter 3 we will see that the normalization map Proposition 1.3.8 is onto but not necessarily one-to-one.
4
3
;
constructed in
Smooth Affine Toric Varieties. Our next goal is to characterize when an affine toric variety is smooth. Since smooth affine varieties are normal (Proposition 1.0.9), 4 3 we need only consider toric varieties coming from strongly convex rational polyhedral cones 6J! ?G@ .
3
We first study when 6 has maximal dimension. Then 6 7 is strongly convex, < 203 R so that 6 7 9 ; has a Hilbert basis . Furthermore, Corollary 1.3.3 tells us 3 . 4 3 4 3 that the torus action on has a unique fixed point, denoted here by . The < 3 point and the Hilbert basis are related as follows. 4
Lemma 1.3.10. Let 6 !?G@ be a strongly convex rational polyhedral cone of 4 4 3 8 be the Zariski tangent space to the affine toric maximal dimension and let 3
! 4 4 3 8 R < 4 3 variety at the above point . Then .
A%>L 3 -
Proof. 1.3.3, the maximal ideal of corresponding to P By Corollary P ,. 23 SX L . Since ' P is a basis of F%>L 3 - , we obtain K'
R ' P R P P(
%' P
irreducible
P
%' P R ' P P
reducible
3
is
R
Chapter 1. Affine Toric Varieties
40
It follows that local ring
!
R
4 4 ( 8
Example 1.3.17. In Example 1.3.7 the dual of 6 R interacts ! with the lattices shown in Figure 10 on the next page. To make this precise, let us name the lattices involved: the lattices Z
R ? ! ? R 4 * .8 * . * I Z have 6J! ? @ ! ? @ , and the dual lattices Z
R ; ; R 4 * 8 * . * I Z have 6 7 !; @ ! ;V@ . Note that duality reverses inclusions and that ; and ? are indeed dual under dot product. In Figure 10 (a), the black dots in the first quadrant 2 34 5 R Z , and in Figure 10 (b), the white dots in the 7 form the semigroup 6 9 ; 2+3 5 R first quadrant form 6 7 9-; .
Chapter 1. Affine Toric Varieties
44
σ
σ
(a)
(b)
Figure 10. Lattice points of
relative to two lattices
and
3 5
4
4
34 5
4
34 5 R
This gives the affine toric varieties . Clearly since Z 4 3 5 is smooth for ? , while Example 1.3.7 shows that is the rational normal Z !? gives a toric morphism cone . The inclusion ?
6
R
4
34 5 R
3 5 R
4
Our next task is to find a nice description of this map.
Z
Z
!? , where ? has finite index in ? , In general, suppose we have lattices ? Z and let 6 ! ? @ R ?@ be a strongly convex rational polyhedral cone. Then the Z ! ? gives the toric morphism inclusion ?
3 5 R Z ; , so that The dual lattices satisfy ; 4
4
34 5
A% 6 7 9 ; Z -
corresponds to the inclusion
&% 6 7 9 ; &% - as a ring of invariants of of semigroup algebras. The idea is to realize 6 7 9; A % Z a group action on 6 7 9-; . Z Z Proposition 1.3.18. Let ? have finite index in ? with quotient R ? ? and Z let 6 ! ? @ R ? @ be a strongly convex rational polyhedral cone. Then:
I. * 4 ; Z ; J M 8 R 4 5 5 8 A% Z - with ring of invariants acts on 6 7 9; A% 6 7 9-; Z - R A% 6 7 9 ; -z 8 . 1.3.12. Let ' and 0 ' be strongly convex rational polyhedral cones. 4 44 This gives the cone ' . Prove that . Also explain 4 how this result applies to (1.3.2). 1.3.13. By Proposition 1.3.1, a point K of an affine toric variety G (C=?7@ AIH J!' is represented by a semigroup homomorphism 6 h . Prove that K lies in the torus of G if and only if never vanishes, i.e., k 'U(V T for all . 1.3.10. Prove the assertions made in the proof of Proposition 1.3.18 concerning the pairing
defined by
+ .
Chapter 1. Affine Toric Varieties
48
Appendix: Tensor Products of Coordinate Rings In this appendix, we will prove the following result used in §1.0 in our discussion of products of affine varieties.
and .
Proposition 1.A.1. If the same is true for
are finitely generated -algebras without nilpotents, then
H d 27+6+7+62 d J!
Proof. Since the tensor product is obviously a finitely generated -algebra, we need only has no nilpotents. If we write prove that , then is radical and hence has a primary decomposition + + , where each + is prime ([17, Ch. 4, §7]). This gives
( v !
IH d 26+7+6+62 d JA #Dh + ! I H d 27+6+6+72 d J!
+ where the second map is injective. Each quotient IH d 26+7+6+92 d JA + is an integral domain q + . This yields an injection and hence injects into its field of fractions 1 M#Dh + ! q + 2
and since tensoring over a field preserves exactness, we get an injection
q
h + ! q
+
+
q
q
Hence it suffices to prove that has no nilpotents when is a finitely generated field extension of . A similar argument using then reduces us to showing that has no nilpotents when and are finitely generated field extensions of . Since has characteristic , the extension has a separating transcendence basis ([56, p. 519]). This means that we can find such that are algebraically independent over and is a finite separable extension. Then
q
c c i c 26+7+6+72 c 26+7+6+92 c c ( 26+6+7+62 ' q Z' Aq I' + q N(Cq I c 62 +7+6+62 c p')(*q c 26+7+6+92 c p' is a field, so that we are reduced But gN(Cq to considering g where g and are extensions of and The latter and the is finite and separable. theorem of the primitive element imply that H JA , ' , where , ' has distinct roots in some extension of . Then g g H JA , ' gH J! , ' + Since , ' has distinct roots, this quotient ring has no nilpotents. Our result follows. A final remark is that we can replace with any perfect field since finitely generated
extensions of perfect fields have separating transcendence bases ([56, p. 519]).
Chapter 2
Projective Toric Varieties
§2.0. Background: Projective Varieties Our discussion assumes that the reader is familiar with the elementary theory of projective varieties, at the level of [17, Ch. 8].
In Chapter 1, we introduced affine toric varieties. In general, a+ toric variety over containing a torus 5 47 M 8 as a Zariski is an irreducible variety + 7 4 8 M open set such that the action of on itself extends to an action on . We will learn in Chapter 3 that the concept of “variety” is somewhat subtle. Hence we will defer the formal definition of toric variety+ until then and instead concentrate on toric varieties that live in projective space , defined by
+ R 47 + U ( S 8 M .4 * * +98 R 4 * * + 8 M + ( where M acts via homotheties, i.e., for . U 4 9 + 8 4 + 8 * + * . * * are homogeneous coordinates of a and . Thus and are well-defined up to homothety. point in
(2.0.1)
)
)
)
)
The goal of this chapter is to use lattice points and polytopes to create toric + varieties that lie in . We will use the affine semigroups and polyhedral cones introduced in Chapter 1 to describe the local structure of these varieties.
+ ; Homogeneous Coordinate Rings. A projective variety ! is defined by the vanishing of finitely many homogeneous polynomials in the polynomial ring L R F% ' *' +.- . The homogeneous coordinate ring of ; is the quotient ring A%>;S- R L :.4; - inherits a grading where ring
%>; -? R VL ? :.4;- , so that we get pieces as follows. The variable ' induces an element '#. the localization 4
'
F%>; - R O ' ' O . A%>;S-;S- has a well-defined -grading as in Exercises 1.0.2 and 1.0.3. Note that
V4 ' ' 8 R V4 /8 R O O given by when O is homogeneous. Then 7 4 A > % S ; - 8 R ' & (2.0.6)
O ' . %>;- O is homogeneous of degree F%>; - consisting of all elements of degree . This gives an affine is the subring of ; piece of as follows.
(2.0.5)
Chapter 2. Projective Toric Varieties
52
Lemma 2.0.3. The affine piece
; 9
F%>; 9
'
4
of
;
has coordinate ring
' - 47A%>; - 8
4
Proof. We have an exact sequence
%>; -NR R :.4;S- R (2.0.7) R :.4 4 $ 8 ! ;J@ Proposition 2.1.9. Given and set R . , is a vertex of . Then .3 R .3 9 4 $
.3 9
9 ' ! Proof. We will prove that if .
, then . . The discussion of polytopes from §2.2 below implies that
9;0/ R
&
3
4
, . 9 &
for some
4
R
Given .
, we have , ' . 9 ; , so that , ' is a convex , combination of the vertices . Clearing denominators, we get integers W such that
-linear and
,-' R & , & R R L 4, R , '8 R Thus & P' " P A% 2 ,- which implies , A' % 2 - , . ' when 2 ' - . Fix A % R . ' ' such a . Then so . By (2.1.6), 3 9 4 ' 9 4 R BC D47A% 2 ' - 8 R is invertible, 3 9 4 ' , giving 3 9 4 ' ! 3 9 4 .
;
+
Projective Normality. An+ irreducible variety ! is called projectively normal U ( ; if its affine cone is normal. A projectively normal variety is always ! normal (Exercise 2.1.5). Here is an example to show that the converse can fail. Example 2.1.10. Let
$ !
consist of the columns of the matrix
. The polytope R = I> 4 $ 8 is the giving the Laurent monomials 4 8 and 4 8 , with vertices corresponding to and . line segment connecting 3 The affine piece of corresponding to has coordinate ring F% - R A% . 4 8 4 8 - R &% -)4 (
Example 2.2.6. Consider the square ( normals of are and and the facet presentation of
* Since the
R L W F K, ( X R L W F K, X
are all equal to , it follows that
lattice polytope. The polytopes
and
.
8 !
=.
of
. The facet is given by
R = I> 4 ( 8
is also a
are pictured in Figure 2.
It is rare that the dual of a lattice polytope is a lattice polytope—this is related to the reflexive polytopes that will be studied later in the book.
= I> 4 ( (
Example 2.2.7. The -simplex R in Example 2.2.4 has facet presentation
K , 4R
K, K, (R
R
L W K, X ( R LXW R LXW LXW
8 !
pictured
R
R
(Exercise 2.2.3). However, if we replace in the last inequality, we get 4 .8 , whichwith integer inequalities that define is not a lattice polytope.
The combinatorial type of a polytope is an interesting object of study. This leads to the question “Is every polytope combinatorially equivalent to a lattice polytope?” If the given polytope is simplicial, the answer is “yes”—just wiggle the vertices to make them rational and clear denominators to get a lattice polytope. The same argument works for simple polytopes by wiggling the facet normals. This will enable us to prove results about arbitrary simplicial or simple polytopes
Chapter 2. Projective Toric Varieties
68
using toric varieties. But in general, the answer is “no”—there exist polytopes in every dimension W not combinatorially equivalent to any lattice polytope. An example is described in [103, Ex. 6.21]. Normal Polytopes. The connection between lattice polytopes and toric varieties comes from the lattice points of the polytope. Unfortunately, a lattice polytope might not have enough lattice points. The -simplex from Example 2.2.7 has only four lattice points (its vertices), which implies that the projective toric variety $ * is just (Exercise 2.2.3).
We will explore two notions of what it means for a lattice polytope to “have enough lattice points.” Here is the first.
.
for all
4 A8 9 ;
.
is normal if
4 A8 9-; R 4*4
4 A 8
!; @
Definition 2.2.8. A lattice polytope
4 A8
8A8
4*4
9;
8A8
9; 9-; 4 ! 8 9-; is automatic. Thus The inclusion come from lattice points of normality means that all lattice points of and . In particular, a lattice polytope is normal if and only if
9-;
9-; R 4 A8 9 ;
times
for all integers W . In other words, normality says that has enough lattice points to generate the lattice points in all integer multiples of .
Lattice polytopes of dimension class of normal polytopes. Definition 2.2.9. A simplex R , , for vertices! , differences ,
are normal (Exercise 2.2.4). Here is another
is basic if has a vertex , such that the , of , form a subset of a -basis of ; .
;@
RP
, . is chosen. The standard This definition + + is independent of which vertex simplex is basic, and any basic simplex is normal (Exercise 2.2.5). More ! general simplicies, however, need not be normal. = I> )4 ( (
Example 2.2.10. Let R that the only lattice points ( of
(
R
4 8
8 !
. We noted earlier are its vertices. It follows easily that ( (
( 4 (8
is not the sum of lattice points of is not basic.
4 8
4 (
. This shows that
8
.
is not normal. In particular,
Here is an important result on normality. Theorem 2.2.11. Let !; @ be a full dimensional lattice polytope of dimension / W . Then is normal for all W / R .
§2.2. Lattice Points and Polytopes
69
Proof. This result was first explicitly stated in [12], though (as noted in [12]), its essential content follows from [28] and [64]. We will use ideas from [64] and [78] to show that 4 A8
8A8 9-; 9 ; 9-; R 4*4 R for all integers W / . In Exercise 2.2.6 you will prove that (2.2.3) implies that W is normal when / R . Note also that for (2.2.3), it suffices to show 4*4 8 A8 9 ; ! 4 A8 9-; 9;
(2.2.3)
since the other inclusion is obvious.
First consider the case where is a simplex with no interior lattice points. Let , + and take W / R . Then 4 8 has vertices the vertices of be , 4 8 , 4 8 , + , so that 4*4 8A8 9 ; is a convex a point , . combination
+ 4 8 + ' , ' where ' W & '( ' R ) 4 8 ' , then If we set ' R + + , R & '( ) ',-' where ) ' W & ')( ) ' R )
for some , then one easily sees that , R , ' . 4 &8 9 ; . Hence If ' W , R 4 , R ,-' 8 ,-' is the desired decomposition. On the other hand, if ) ' , R & '(
for all , then
+ / R 4 / +R 8 R & '( ) ' / R and & ')( ) ' R / . Now consider the lattice point so that R / + , R , , + 8 R , R & ')( 4 R ) ' 8 ,-' + 4 R ) 8 )
' ' R The coefficients are positive since for all , and their sum is & '(
R
0) R R , ' / / for . Hence is a lattice point in the interior of since
all . This contradicts our assumption on and completes the proof when lattice simplex containing no interior lattice points.
is a
To prove (2.2.3) for the general case, it suffices to prove that is a finite union of / -dimensional lattice simplices with no interior lattice points (Exercise 2.2.7). For this, we use Carath´eodory’s theorem (see [103, Prop. 1.15]), which asserts that $ !; @ , we have for a finite set
= I> ) 4 $ 8 R
$
= I> ) 4 8
! = I>)4 $ 8
where the union is over all subsets of affinely = I>! 4 consisting 8 is a simplex. independent elements. Thus each This enables us to write our lattice polytope as a finite union of / -dimensional lattice simplices.
Chapter 2. Projective Toric Varieties
70 If an / -dimensional lattice simplex lattice point , then +
#
R
' ' ( #
#
R = I> 4 + 8 #
has an interior
' R = I> 4 ' +2 8
is a finite union of / -dimensional lattice simplices, each of which has fewer interior lattice points than # since becomes a vertex of each # ' . By repeating this process on those # ' that still have interior lattice points, we can eventually write # and hence our original polytope as a finite union of / -dimensional lattice simplices with no interior lattice points. You will verify the details in Exercise 2.2.7.
This theorem shows that for the non-normal -simplex of Example 2.2.10, is normal. Here is another consequence of Theorem 2.2.11. its multiple Corollary 2.2.12. Every lattice polygon
!
is normal.
We can also interpret normality in terms of the cone
4 A8 R = I> 4 8 !;J@
introduced in Figure 3 of Chapter 1. The key feature of this cone is that is the 4 A8 at height , as illustrated in Figure 4. It follows that lattice points “slice” of ,. correspond to points 4 , 8 . 4 A8 9 4 ; X8 .
2P
height = 2
C(P) P
Figure 4. The cone
height = 1
sliced at heights
In Exercise 2.2.8 you will show that the semigroup 4 &8 points in relates to normality as follows.
and
4 A8 9 4 ; X8
of lattice
§2.2. Lattice Points and Polytopes
71
Lemma 2.2.13. Let ! ; @ be a lattice polytope. Then 4 9-; 8X generates the semigroup 4 A8 9 4 ; if
X8is. normal if and only ! ; X8 @ is normal if and only if 4 9 ; 8 is This lemma tells us that 4 A8 4 the Hilbert basis of . 9 ; = >I )4 ( ( Example 2.2.14. In Example 2.2.10, the simplex R 8 gives the cone 4 A8 ! . The Hilbert basis of 4 A8 9 4 ; X8 is 4 8 4 ( 8 4 8 4 ( 8 4 ( .8 4 ( .8
(Exercise 2.2.3). Since the Hilbert basis has generators of height greater than ,
Lemma 2.2.13 gives another proof that
is not normal.
+
In Exercise 2.2.9, you will generalize Lemma 2.2.13 as follows.
be a lattice polytope of dimension / W and Lemma 2.2.15. Let ! ; @ 4 &8 let be the maximum height of an element of the Hilbert basis of . Then:
(a) (b)
/ R .
is normal for any
W
.
The Hilbert basis of the simplex of Example 2.2.14 has maximum height . is normal. The paper [64] gives a Then Lemma 2.2.15 gives another proof that version of Lemma 2.2.15 that applies to Hilbert bases of more general cones. Very Ample Polytopes. Here is a slightly different notion of what it means for a polytope to have enough lattice points.
! ;@ is very ample if for every vertex Definition 2.2.16. A lattice polytope , . , the semigroup 2 $ P R &4 9J; R , 8 generated by 9J; R , R
, Z R , , Z . 9-; is saturated in ; .
This definition relates to normal polytopes as follows. Proposition 2.2.17. A normal lattice polytope
. Proof. Fix a vertex ,
is very ample.
, . ; such that , . 2 $ P for some 2 $ P
, . 2 $ P as integer W . To prove that ,. , write , R & P $ * P 4 , Z R , 8 * P . F so that @ W & P $ * P . Then Pick @ . , @ , R & P $ * P , Z @ R & P $ * P , . @ Dividing by gives , @ , . @ , which by normality implies that
and take
@ , R ' ( ? 4 R , ' & ( ( ,-' We conclude that ,
?
, ' . 9; for all ( ,-' R , 8 . 2 $ P , as desired.
Chapter 2. Projective Toric Varieties
72
+ ! ; @
Combining this with Theorem 2.2.11 and Corollary 2.2.12 gives the following. Corollary 2.2.18. Let
! W (a) If , then
! R (b) If , then
be a full dimensional lattice polytope.
is very ample for all
W / R
.
is very ample.
Part (a) was first proved in [28]. We will soon see that very ampleness is precisely the property needed to define the toric variety of a lattice polytope. The following example taken from [11, Ex. 5.1] shows that very ample polytopes need not be normal, i.e., the converse of Proposition 2.2.17 is false.
% R
Example 2.2.19. Given
in positions and elsewhere. % , letThus $ R % - 4 , ' is a vertex, it has a supporting Proof. Let ' R 9V; O R U , ' 8 . Since O O R ,-' . It follows that O is hyperplane such that ! and 9 ' (Exercise 2.3.1), so that ' is strongly convex. a supporting hyperplane of .
! ' R ! It is also easy to see that (Exercise 2.3.1). It follows that ' and 6 ' R ' 7 are strongly convex rational polyhedral cones of dimension / . 2 ' 9V; R 6 ' 7 9V; . By hypothesis, is very ample, which We have ' ! 2 2 means that ' ! ; is saturated. Since ' and ' R 6 ' 7 are both generated by where = cone
Chapter 2. Projective Toric Varieties
76
2 R ,-' , part (a) of Exercise 1.3.4 implies ' R 6' 7 9 ;
9 ;
. (This exercise was part of the proof of the characterization of normal affine toric varieties given in Theorem 1.3.5.) Part (a) of the theorem follows immediately.
' and
For part (b), Theorem 1.2.18 implies that 4 5 ! 4 3 R $ strongly convex. Then 9 $ . the torus of
5 is the torus of 4 3 since 6 ' is 5 is also ' ! $ shows that
$
The affine pieces intersect in 9 9 9 ' 9 order to describe this intersection carefully, we need to study how the cones 6 6 fit together in ?G@ . This leads to our next topic.
$
4
$
4
4
4
'
. In and
The Normal Fan. The cones 6 ' ! ? @ appearing in Theorem 2.3.1 fit together in a remarkably nice way, giving a structure called the normal fan of . Let !; @ be a full dimensional lattice polytope, not necessarily very ample. Faces, facets and vertices of will be denoted by # , and respectively. Hence we write the facet presentation of as
R , . ;J@ K , LXW R *
(2.3.1)
A vertex
(When R containing
.
gives cones
R = I> 4 9 ; R 8 ! ;V@
and correspond bijectively to faces Q R+ Q R = I> 4 # #
(2.3.2)
Q
R+
R 4Q
#
6 R
and
, ' , these are the cones '
!
89
6 '
9-;
.
for all
R
7
! ? @
studied above.) Faces via the maps
! #
8
which are inverses of each other. These maps preserve dimensions, inclusions, and intersections (Exercise 2.3.2), as illustrated in Figure 5 on the next page. In particular, all facets of
come from facets of
R , . ;V@ K , LXW
for all
containing
By the duality results of Chapter 1, it follows that the dual cone 6
= 6 R I> 4
6"!
R = I> 4
8
contains
This construction generalizes to arbitrary faces
! #
containing , so that
.
is given by
by setting
contains
#
8
R since is the cone Thus the cone 6 is the ray generated by , and 6 generated by the empty set. Here is our main result about these cones. Theorem 2.3.2. Let
6"! # is a face of (a) For all 6
!
. Y%$
(b) The intersection
$
; @ be a full dimensional lattice polytope and set !. Then:
, each face of 6
6 ! 9 6
!
Y%$
of any two cones in Y . !
is also in
$
is a face of each.
Y
$
R
§2.3. Polytopes and Projective Toric Varieties
77
P Cv
v 0
Figure 5. The cone
of a vertex
A collection of strongly convex rational polyhedral cones satisfying conditions (a) and (b) of Theorem 2.3.2 is called a fan. General fans will be introduced in Y $ are built from the inward-pointing Chapter 3. Since the cones in the above fan % Y $ normal vectors , we call the normal fan or inner normal fan of .
The following easy lemma will be useful in the proof of Theorem 2.3.2.
O be a supporting affine hyperplane ! O 9 . ) , ) W . Then Proof. First suppose that . 6 ! and write R & ! R & ) * easily implies that ! O U and # ! .O 9 . setting R ! * come from the facet presentation (2.3.1). Recall that the integers O 9 . Take a vertex . # . Then Going the other way, suppose that # ! U U O O O . 7 R 6 , so that ! and imply that ! . Hence . R & ) ) W Let be a facet of containing but not # , and pick . # with . . Then . # ! . O imply that R K L R & ) K L R K L R & ) K L R R & ) * R * for . . These equations imply where the last equality uses K L R & ) K L R R & ) *
Lemma 2.3.3. Let # be a face of of . Then . 6"! if and only if #
and let
Chapter 2. Projective Toric Varieties
78
However, . gives K L ) R follows that whenever proof of the lemma. Corollary 2.3.4. If
#
is a face of #
, and since K AL W R * for all , it ! P . This gives . 6 and completes the . 6 if and only if ! . , then R *
!
"!
#
Proof. One direction is obvious by the definition of 6 ! , and the other direction O " follows from Lemma 2.3.3 since is a supporting affine hyperplane of " O with 9 R .
Theorem 2.3.2 is an immediate corollary of the the following proposition. Proposition 2.3.5. Let ! ; @ . Then:
!
Z
and #
if and only if 6
Z
#
!
be faces of a full dimensional lattice polytope
6"!
Z (b) If # ! # , then 6 ! is a face of 6"! , and all faces of 6 ! are of this form. ZZ Z (c) 6"!9 6 ! R 6 ! , where # is the smallest face of containing # and # . Z Z Proof. To prove part (a), note that if # ! # , then any facet containing # also (a)
#
#
!
.
contains # , which implies 6 ! ! 6! . The other direction follows easily from Corollary 2.3.4 since every face is the intersection of the facets containing it by Proposition 2.2.1.
For part (b), fix a vertex . # and note that by (2.3.2), # determines a face Q of . Using the duality of Proposition 1.2.10, Q gives the dual face
QM
of the cone obtains
6
. Then using 6
7
QI
9
R = I> 4
R = I> 4
QM
R
O
R 6 9
.
Q
.
QI
8
! O
and Q
Since . # , the inclusion Q ! is equivalent to turn is equivalent to # ! . It follows that
QM
(2.3.3)
Z
R = I> 4
#
! O
R 6 7
"
, one
, which in
8R 6
!
#
8
!
"!
arise in this way. ! 6 In particular, ! means that 6 is also a face of 6 , and since 6 by part (a), we see that 6 a face of 6 . Furthermore, every face of 6 a face for some face isZ . Using of 6 by Proposition 1.2.6 and hence is of the form 6 so that 6"! is a face of 6 , and all faces of 6 #
-#
!
!
!
part (a) again, we see that
ZZ
!
#
!
!
Z , and part (b) follows.
#
!
#
!
Z
For part (c), let # be the smallest face of containing # and # . This exists Z Z is the interbecause a face is the intersection of the facets containing it, so that # Z Z Z R ). section of all facets containing # and # (if there are no such facets, then # By part (b) 6 ! is a facet of both 6 ! and 6 ! . Thus 6 ! ! 6"!9 6 ! .
§2.3. Polytopes and Projective Toric Varieties
79
It remains to prove the opposite inclusion. If 6 ! 9V6 ! R R 6 , then Z Z R and we are done. If 6 ! 9 6 ! R P , any nonzero in the intersection # lies in both 6"! and 6 ! . The proof of Proposition 2.3.6 given below will show that O. . By Lemma 2.3.3, is a supporting affine hyperplane of for some . . 6"! and . 6 ! imply that # and # Z lie in O. 9 . The latter is a face of Z Z Z ! O. 9 since # Z Z is the smallest such face. containing # and # , so that # Applying Lemma 2.3.3 again, we see that . 6 ! .
$
Proposition 2.3.5 shows that there is a bijective correspondence between faces Y $ of and cones of the normal fan . Here are some further properties of this correspondence. Proposition 2.3.6. Let ! ; sion / and consider the cones
(a) (b)
!
? @ R
#
@
be a full dimensional lattice polytope of dimenY $ of . Then: in the normal fan
6 !
! R 6 ! / for all faces # " $ R 3 6 6! 6 vertex of
of
.
.
Proof. Let # be a face of and take a vertex of # . By (2.3.2) this gives a face Q of the cone , which has a dual face Q M of the dual cone 7 R 6 . Since QM R 6"! by (2.3.3), we have
!
#
! 6
!
R
Q
!
R /
QM
where the first equality uses Exercise 2.3.2 and the second follows from Proposition 1.2.10. This part (a). For part (b), U let . ? @ be nonzero and set K L proves = O and . O for at least one =R vertex of . Then ! vertex of , so that . 6 by Lemma 2.3.3. The final equality of part (b) follows immediately.
A fan satisfying the condition of part (b) of Proposition 2.3.6 is called complete. Thus the normal fan of a lattice polytope is always complete. We will learn more about complete fans in Chapter 3. An important observation is that the normal fan in Example 2.3.9 doesn’t de pend on the integer . In general, multiplying a polytope by a positive integer has no effect on its normal fan, and the same is true for translations by lattice points. We record these properties in the following proposition (Exercise 2.3.3). Proposition 2.3.7. Let ! ; @ be a full dimensional lattice polytope. Then for
any lattice point ,. ; and any integer W , the polytopes , and have the same normal fan as .
Examples of Normal Fans. Here are some examples of normal fans.
Example 2.3.8. Figure 6 on the next page shows a lattice hexagon in the plane ( , with corre together with its normal fan. The vertices of are labeled ( 6 in the normal fan. In the figure, is shown on the left, sponding cone 6
Chapter 2. Projective Toric Varieties
80
and at each vertex ' , we have drawn the normal vectors of the facets containing ' and shaded the cone 6 ' they generate. On the right, these cones are assembled at the origin to give the normal fan.
v5
v4 σ3
σ2
v6 v3
v1
σ4
ΣP
P
σ1 σ5
v2 Figure 6. A lattice hexagon
and its normal fan
σ6
Notice how one can read off the structure of from the normal fan. For Y $ example, two cones 6 ' and 6 share a ray in if and only if the vertices ' and lie on an edge of .
(
Example 2.3.9. The -simplex for ! has vertices Y . $Let R some positive integer . Figure 7 shows and its normal fan . At each vertex
v2 ΣP
P
v0
σ0 σ2
v1
Figure 7. The triangle
' of
σ1
and its normal fan
, we have drawn the normal vectors of the facets containing ' and shaded Y $ the cone 6 ' they generate. The reassembled cones appear on the left as .
Here is an example of the normal fan of a polytope in
.
4 8 .
Example 2.3.10. Consider the cube with vertices ( , and the ! facet presentation facet normals are of
K , ' LXW R
is
The
§2.3. Polytopes and Projective Toric Varieties
81
The origin is an interior point of . By Exercise 2.2.1, the facet normals are the vertices of the dual polytope , the octahedron pictured in Figure 8. z
z
y
y
P
P˚ x
x
Figure 8. A cube
and its dual octahedron
However, the facet normals also give the normal fan of , and one can check that in the above figure, the maximal cones of the normal fan are the octants of , which are just the cones over the facets of the dual polytope .
As noted earlier, it is rare that both and are lattice polytopes. However, !;J@ is a lattice polytope containing whenever an interior point, it is Y $ are theascones over the facets of still true that maximal cones of the normal fan ! ? @ (Exercise 2.3.4).
.
The special behavior of the polytopes and and 2.3.10 leads to the following definition.
discussed in Examples 2.2.6
Definition 2.3.11. A full dimensional lattice polytope facet presentation is
R , . ;J@ K , L W R
If is reflexive, then is a lattice point of * R for all point of (Exercise 2.3.5). Since
Thus
R = I> ) 4
! ;G@
for all facets
is reflexive if its
.
and is the only interior lattice , Exercise 2.2.1 implies that
facet of
A8
is a lattice polytope and is in fact reflexive (Exercise 2.3.5).
Chapter 2. Projective Toric Varieties
82
We will see later that reflexive polytopes lead to some very interesting toric varieties that are important for mirror symmetry.
$
The General Case. Earlier in the section, we studied the toric variety of a $ . This development very ample polytope and described the affine pieces of leaves one item of unfinished business: What is the intersection of two + of the affine pieces described in Theorem 2.3.1?
To study let !; R 9-this, ; . Then
@
be an / -dimensional very ample polytope and set
"" ( ! "" ( ' P Label the homogeneous as for ,. 9 ; . If is a vertex ( coordinates of 8 " ' " of , then ! is the affine open subset where R P , and Theorem 2.3.1
$
4
3 R B9C D47&% 7 6 9 ; -8 $ 9 4 is the affine toric variety of the cone 6 in the normal fan Y $ of So Proposition 2.3.12. Let ! ; @ be full dimensional and very ample. If R P $
tells us that
9
4
R
4
9 9 R 3 R and the inclusions 9 9 9 are vertices of
and
#
is the smallest face of 4
$
3 4
(2.3.4)
4
4
$
can be written
4
4
$
4 4 3 8
R
containing
and
BC D47A% 6 7 9; - 8 ! 4
!
3 R 4 4 3 8 4
!
, then
9 4
$
.
3
4
$
Proof. We analyzed the intersection of affine pieces of in §2.1. Translated to the notation being used here, (2.1.6) and (2.1.7) imply that
$
9 4
3 9 4 R 4 4 8
R 44 3 8
4 4 3 8 R 4 3 R . R 6 7 , so that H R O " 96 is a face of 6 . In However, we have this situation, Proposition 1.3.16 and equation (1.3.4) imply that 44 3 8 R 4 O " 9 6 R 6 ! . Thus the proposition will follow once we prove H R 6 ! , i.e., Since 6"! R 6 9 6 by Proposition 2.3.5, it suffices to prove that O " 9 6 R 6 9 6 O " 9V6 O Let . . If R P , there is . is a supporting affine O such O Thus all we need to show is that
hyperplane of . Then . 6 implies . by Lemma 2.3.3, so that . O " . Applying Lemma 2.3.3 again, we get . 6 . Going the other since .
§2.3. Polytopes and Projective Toric Varieties
83
R P , pick . as above. Then . 6 9V6 and . O , from which . O " follows easily. This
way, let . 6 9 6 . If Lemma 2.3.3 imply that completes the proof.
This proposition and Theorem 2.3.1 have the remarkable result that the normal Y $ $ $ fan completely determines the internal structure of : we build 4 3 from local pieces given by the affine toric varieties( , glued together via (2.3.4). "" for any of this—everything we We don’t need the ambient projective space need to know is contained in the normal fan.
As a consequence, we can now give the general definition of the toric variety of a polytope. Definition 2.3.13. Let define
where
! ;@
be a full dimensional lattice polytope. Then we
$ R
$
is any positive integer such that
is very ample.
Such integers exist by Corollary 2.2.18, and if and are two such integers, Y $ R then and have the same normal fan by Proposition 2.3.7, namely Y $ R Y $ $ $ . It follows that while and lie in different projective 4 3 glued together via (2.3.4). spaces, they are built from the affine toric varieties Once we develop the language of abstract varieties in Chapter 3, we will see that $ is well-defined as an abstract variety.
$
We will often speak of without regard to the projective embedding. When $ we want to use a specific embedding, we will say “ is embedded using ”, is very ample. In Chapter 6 we will use the language of where we assume that $ such that $ divisors and line bundles to restate this in terms of a divisor on $ is very ample precisely when is.
Here is a simple example to illustrate the difference between $ as sitting in a specific projective space. variety and
$
as an abstract
+
+
! Example 2.3.14. Consider the / -simplex . We can define using + + + W
for any integer since and hence very ample. The lattice U + R is normal F% ( +.- of total monomials points in correspond to the of ( + + " "
R R degree / . This gives an embedding . When , ! 9 ( , +
implies that +
R
The normal fan of ! can regard
+ in Exercise 2.3.6. 8" " is( described as the image of the map $ + R " " (
For an arbitrary
W , we
Chapter 2. Projective Toric Varieties
84
F% ' *' + -
defined using all monomials of total degree in (Exercise 2.3.6). It follows that this map is an embedding, usually called the Veronese embedding. But + when we forget the embedding, the underlying toric variety is just .
The Veronese embedding allows us to construct some interesting affine open + F% ' *' + - be nonzero and homogeneous of degree( subsets of . Let O . ' . We write the homogeneous coordinates of " " as and write O R & ( " R for . Then R & ( is a nonzero linear form in the variables " " ( /" " ( S 354 8 , so that is a copy of (Exercise 2.3.6). If follows that
+ S 5 3 4 O 8 $ 4 + 8 9 " " ( S 354 8 + has a richer supply of is an affine variety (usually not toric). This shows+ that S 354 ' ' 8 considered earlier in affine open subsets than just the open sets ' R 4
the chapter.
+ later in the + book, we will see the When we explain the construction of + Proj intrinsic reason why S 354 O 8 is an affine open subset of . + Example 2.3.15. The -dimensional analog of the rational normal curve is the L rational normal scroll , which is the toric variety of the polygon = >I 4 * ( ( 8 R !
* * and its normal fan are where . satisfy . The polygon R
pictured in Figure 9.
v2 = e2 P = P2,4 v1 = 0
σ4
v3 = 4e1 + e2
σ1
ΣP
σ3 σ2
v4 = 2e1
Figure 9. The polygon of a rational normal scroll and its normal fan
*
lattice points and gives the map 47 M 8 +R U U ( 4 8 4 8 !L R $ is the Zariski closure of the image. To describe the image, such that we rewrite the map as ( R+ U U ( 4 . )) 8 4 )) )) ) ) 8 In general, the polygon
has
§2.3. Polytopes and Projective Toric Varieties
4 ) 8
4 8
85
4 8
R , the map is , which When is the * coordinates of U U ( . In the rational normal curve mapped to the first 4 ) 8 R 4 8 gives same way, ( the rational normal curve mapped to the last
U U coordinates of . If we think of these two curves as the “edges” of a 4 )) 8 . ( vary gives scroll, then fixing gives a point on each edge, and letting the line of the scroll connecting the two points. So it really is a scroll!
important observation is that the normal fan depends only on the difference R An * , since this determines the slope of the slanted edge of . If we denote the , it follows that as abstract toric varieties, we have difference by . $ R $ R $ R
since they are all constructed from the same normal fan. In Chapter 3, we will see < that this is the Hirzebruch surface . (
L
U U
*
But if we think of the projective surface ! , then and have a unique meaning. For example, they have a strong influence defining equa U U ( ' onthe L * ' tions of . Let the homogeneous coordinates of be 4* 8 and consider the matrix
!
' ')( ' " ( ( " ( )' ( ' ' ( 624;S- and O ' 4
"
This completes the proof. Here is a consequence of Proposition 3.0.12 and Definition 3.0.11.
Proposition 3.0.13. Let ; affine open sets . Then
be an irreducible variety with a cover consisting of ; is normal. is normal if and only if each
Products of Varieties. As another example of abstract varieties and gluing, we ( V of varieties ( and also has the structure indicate why the product of a variety. In §1.0 we constructed the product of affine varieties. From here, it ( is relatively routine to see that if is obtained by gluing together affine varieties 4 is obtained by gluing together 4 Z ( is obtained by and affines , then 4 4 Z ( - in the corresponding fashion. Furthermore, gluing together the has the correct universal mapping property. Namely, given a diagram
$ ( %
( #
/
' are morphisms, there is a unique morphism where ' (the dotted arrow) that makes the diagram commute. ( (
=
$
R Example 3.0.14. Let us construct the product ; R BC D.47F% - 8 and ;( R BC D47A% - 8 , with the gluing. Write given by
( Then
A% -
is constructed from 4
A%
-
(
; ;V(
where
R BC D.47A% - &&% ' - 8
H( R BC D.47A% -1A&% ' - 8 4
with gluing given by
44 8 44 ( 8
corresponding to the obvious isomorphism of coordinate rings.
Chapter 3. Normal Toric Varieties
102
Separated Varieties. From the point of view of the classical topology, arbitrary gluings can lead to varieties with some strange properties.
( Example 3.0.15. In Example 3.0.14 we saw how to construct affine vari; R BC D47&% - 8 and ;( R BC D.47F% - 8 with thefrom gluing given by eties on open sets M 4[ \ ~ Y [ \ (b) Show that zW{}| \ is a well-defined integer.
K
(c) Deduce that the proposed notion of smoothness at is well-defined.
K% G and set _ \( /&,%% O RK' .
3.0.3. This exercise explores some properties of the morphisms defined in Definition 3.0.3.
' _ \'
(a) Prove the claim made in Example 3.0.4. Hint: Take . Then describe in terms of
, RKS')(* `8
(b) Prove the properties of morphisms listed on page 96.
,2n
3.0.4. Let
be an irreducible abstract variety.
, - if , O (- O
M (b) Show that the set of equivalence classes of the relation in part (a) is a field. (c) Show that if M is a nonempty affine open subset of , then I ' I' . (a) Let be rational functions on . Show that open set is an equivalence relation.
for some nonempty
3.0.5. Show that a variety is Hausdorff in the Zariski topology if and only if it consists of finitely many points.
§3.0. Background: Abstract Varieties
105
3.0.6. Consider Proposition 3.0.18. (a) Prove part (a) of the proposition. Hint: that if Show first by , then ' .
6 h
is defined c ')(M , c ' 2nS c 'n' *( 'n' (b) Prove (b) of the proposition. Hint: Show first that . G can be identified with ' part G$'^ . 6eG h G G 3.0.7. Let G ( <W=D?E@mA' be an affine variety. The diagonal mapping corresponds to a -algebra homomorphism h . Which one? Hint: Consider the universal mapping property of G G. $ , the blowup of 3.0.8. In this exercise, we will study an important variety in F 4 ' from Example 3.0.8. Write the at the origin, denoted F ' . This generalizes d d as 26+7+6+92 , and the affine coordinates on as homogeneous coordinates on c 26+7+6+92 c . Let ( A ')( d + c # d c +)O> fe'^ + (3.0.6) : Let 1+ , (N126+7+6+92 , be the standard affine opens in + ( d + ' 2 k (M 1 26+7+6 +92 . (note the slightly non-standard indexing). So the + form a cover of (a) Show that for each P(Mm26+7+6+92 , + ( + ' d <W=D?E@ d + 26+7+6+62 dd ++ 4 2 d d+ + 26+7+6+72 d d + 2 c + using the equations (3.0.6) defining
d ' and % d + ' . 3.0.9. Let G ( c 4$# d ' 4 and consider the morphism 6G h given by projection onto the d -axis. We will study the fibers of . A ' ( / F `23 1' 8 can be represented as the fibered (a) As noted in the text, the fiber d d .
(b) Give the gluing data for identifying the subsets
+
/E `8 G . In terms of coordinate rings, we have /E `8( =D?E@mFIH J! ' , *(C<W=D?E@mA H d J ' and GM(C<W=D?E@mA H d 2 c J! c 4# d . Prove that IH d J! d H d 2 c JA c 4 # d IH c J! c 4 + Thus, the coordinate rings IH d JA d , IH d J and H d 2 c JA c 4 # d lead to a tensor product that has nilpotents and hence cannot be a coordinate ring. T in , then W')(N/> 2 W' 8 . Show that the analogous tensor product is (b) If (* H d J! d # H d 2 c JA c 4 # d H c JA c 4 # H c JA c # IH c JA c + W' . This has no nilpotents and hence is the coordinate ring of What happens in part (a) is that the two square roots coincide, so that we get only one point with “multiplicity ( .” The multiplicity information is recorded in the affine scheme =?7@1AIH c J! c 4 ' . This is an example of the power of schemes. product
Chapter 3. Normal Toric Varieties
106
§3.1. Fans and Normal Toric Varieties
Y
6
In this section we construct the toric variety corresponding to a fan . We will 6 to many of the examples encountered previously, and also relate the varieties we will see how properties of the fan correspond to properties such as smoothness 6 . and compactness of The Toric Variety Associated to a Fan. A toric variety continues to mean the same thing as in Chapters 1 and 2, although we now allow abstract varieties as in §3.0.
Definition + 3.1.1. A toric variety is an irreducible variety a torus 5 47 M 8 as a Zariski open subset such that the action of 5 containing on itself extends 5 on . to an action of The other ingredient in this section is a fan in the vector space
Y A fan
? @
.
in ? @ is a finite collection of cones 6 such that: Y . (a) Every 6 is a strongly convex rational polyhedral cone. Y Y . (b) For all 6 , each face of 6 is also in . ( Y , the intersection 6 ( 9 6 is a face of each (hence also in Y (c) For all 6 6 .
Definition 3.1.2.
).
It is also possible to consider infinite collections of cones with these properties, but we will not do so in this book. We have already seen some examples of fans. Theorem 2.3.2 shows that the Y $ of a full dimensional lattice polytope !; @ is a fan in the sense normal fan of Definition 3.1.2. However, there exist fans that are not equal to the normal fan of any lattice polytope. An example of such a fan will be given in Example 4.2.13. We now show how the cones in any fan give the combinatorial data necessary to glue a collection of affine toric varieties together to yield an abstract toric variety. Y By Theorem 1.2.18, each cone 6 in gives the affine toric variety
3 R BC D.47A% 23 - 8 R BC D47A% 7 6 9; - 8 Moreover, if H is a face of 6 , then by Definition 1.2.5, there is , . 6 7 such that H R 6 9 O P , where O P R . ? @ K , L R is the hyperplane defined by , . In Chapter 1, we proved two useful facts: O P is the corresponding face of 6 , then First, if , . 6 7 9 ; and H R 6-9 4
Proposition 1.3.16 implies that
2 R 23
4FR , 8 % 2 - is the localization A% 2+3 - . Hence 4 R 4 4 3 8 F This shows that ( Second, if H R 6 9 6 , then Lemma 1.2.13 implies that (3.1.2) 6 ( 9 O P R H R 6 9 O PF (3.1.1)
.
§3.1. Fans and Normal Toric Varieties where ,. (3.1.3)
107
67( 9 F4 R 6 8 A 7 9-; . This shows that 4 3 4 4 3 8 R 4 R 4 4 3 8
!
4
3
These facts were also used to study projective toric varieties in Chapter 2. The 2 3 and their semigroup following proposition gives an additional property of the rings that we will need.
(6 . Y
Proposition 3.1.3. If 6
2 3
6 (7 R 4FR 4 6 ( 98 6 8 7 , . 67 9 4FR , 8 for 6 some79 ; . 203 203 that . . 6 (7
Proof. The inclusion
R 6 ( 9 6
and H
2 R 2 3
23
, then
2 3 2 follows directly from the general fact that ! R H 7 . For the reverse inclusion, take . 2 and let ( (3.1.2). Then (3.1.1) applied to 6 gives R 20satisfy 3 . But R , . 6 7 implies R ,. 203 , so and .
This result is sometimes called the Separation Lemma and is a key ingredient 6 are separated using Definition 3.0.16. in showing that the toric varieties
(
Example 3.1.4. Let 6 R = I> 4 ( ( 8 = inI> ? 4@ ( R dual cones 6N7 in Figure 1.
= I> 4 ( 8 1.2.11), and let 6 R ( (as in Exercise = > I 4 ( 8 H R 6 9A6 R R . We show the (4R ( . Then 8 8 R = I> 4 (+R H 7 R 6(7 6 7 67 ,
, and
τ σ1
σ1 τ
σ2
σ2
Figure 1. The cones
The darkest shaded region on the right is 6 6 9 O " P , where , R R ( Z . 6(7 and R , with , . 6 equals the set of all sums , 2 R 23 23 .
and their duals
( O P R (7 have H R 6 9 9 6 7 .( We 2 R , R Z . 67 . Note that (7 9 ; and , . 6 7 9 ; . Hence
Chapter 3. Normal Toric Varieties
108
4
6 H
BC D47A% 23 - 8
3
R Now consider the collection of affine toric varieties , where ( Y runs over all cones in a fan . Let 6 and 6 be any two of these cones and let R 6 ( 9 6 . By (3.1.3), we have an isomorphism
4
3 3
4 38 4
44 38
which is the identity on . By Exercise 3.1.1, the compatibility conditions as in 4 3 4 4 3 8 are satisfied. §3.0 for gluing the affine varieties 6 along the subvarieties Y associated to the fan . Hence we obtain an abstract variety Theorem 3.1.5. Let variety.
Y
be a fan in ?
Y
@
. The variety
6
is a normal separated toric
Y
. . Proof. Since each cone in is strongly convex, 5 R BC D.47A% - 8 47 M 8 + 4 3 ! ? is a face of all 6 Hence we have for all 6 . These tori are all ; 5 6 ! identified by the gluing, so we have ! . We know from Chapter 1 that each 4 3 3 3 5 has an action of reduces to the identity 203 3 - . The gluing isomorphism & % mapping on . Hence the actions are compatible on the intersections of every pair of sets in the open affine cover, and patch together to give an action of 5 on 6 .
6
4
3
The variety is irreducible because all of the irreducible affine toric 5 . Furthermore, 4 3 is aarenormal varieties containing the torus affine variety by 6 is normal by Proposition 3.0.13. Theorem 1.3.5. Hence the variety
6
To see that is separated it will suffice to show that for each pair of cones 6 ( 6 in Y , the image of the diagonal map 4 4 3 4 3 H R ( 6 9-6 is Zariski closed (Exercise 3.1.2). But comes from the -algebra homomor-
M A% 23 - 1=F% 23 -8R+ &% 2 + ' ' ' P U + defined by . By Proposition 3.1.3, M is surjective, so that A% 2 - 74 A% 203 - &&% 23 - 8 4 M 8 ; 3 ; 3. Hence the image of is a Zariski closed subset of phism
P
6
Toric varieties were originally known as torus embeddings, and the variety 5 4 Y 8 in older references such as [76]. Other commonly would be written 4 Y 8 , or Q4 8 , if the fan is denoted by . When we want to used notations are 6 as 6 5 . emphasize the dependence on the lattice ? , we will write
Y
Every normal, separated toric variety is obtained as for some fan in ?G@ . 6 for For example, Theorem 1.3.5 implies that every normal affine toric variety is the fan consisting of a single cone 6 together with all of its faces. The construction of the projective toric variety associated to a lattice polytope in Chapter 2 shows 6 for suitable fans Y . that those varieties are also 6
§3.1. Fans and Normal Toric Varieties
Proposition 3.1.6. Let the projective toric variety
6
$
109
+ ! ; @ be an / -dimensional lattice polytope. Then Y $ is the normal fan of . , where
Proof. When is very ample, this follows immediately from the description of $ in Proposition 2.3.12 and the the intersections of the affine open pieces of Y $ definition of the normal fan . The general case follows since the normal fans of and are the same for all positive integers . The general statement is a consequence of a theorem of Sumihiro from [98].
5
Theorem 3.1.7 (Sumihiro). Let the torus act on a normal separated variety has a 5 -invariant affine open neighborhood. Then every point .
.
Corollary 3.1.8. Let be a normal separated toric variety containing the torus 5 as an affine open subset. Then there exists a fan Y in ? @ such that 6 . Proof. The proof will be sketched in Exercise 3.2.10 after we have developed some 5 -orbits on toric varieties. of the properties of Additional Examples. We now turn to some concrete examples. Many of these are toric varieties already encountered in previous chapters.
Y
Example 3.1.9. Consider the fan in ?A@ R in Figure 2, where ? R has ( standard basis . This is the same as the normal fan of each of the simplices R as in Example 2.3.9. Here we show all points in the cones inside a rectangular viewing box (all figures of fans in the plane in this chapter will be drawn using the same convention.)
σ1
σ0
σ2
Figure 2. The fan
for
From the discussion in Chapter 2, we expect , and we will show Y R = I> 4 ( 8 , this in detail. The fan has three two-dimensional cones 6 = = 6 ( R I> 4FR ( R 8 , and 6 R I> 4 (4R (R 8 , together with the three 6
Chapter 3. Normal Toric Varieties
110
rays H ' R 6 ' 9 6 for R P , and the origin. We see that the toric variety covered by the affine opens
3 R B C D47&% 2 3 - 8 BC D47A% ' 3 R BC D47&% 23 - 8 B C D47A% ' 3 R BC D47&% 23 - 8 B C D47A% '
4 4 4
6
is
-8 " ( *' " ( - 8 " ( " (-8
%' - & % ' " ( * ' " ( - M( F % ' - & %' " ( " ( - M &
Moreover, by Proposition 3.1.3, the gluing data on the coordinate rings is given by
( ( - &% ' *' " ( - M ( A% ' " *' "
It is easy to see that if we use the usual homogeneous coordinates ' and identifies the standard affine open , then 6 6 4 3 ! . Hence we have recovered as the toric variety .
+ Example 3.1.10. Generalizing Example 3.1.9, let ? @ R ( +
standard basis
. Set
Y
R R ( R % R
4 '*')(*' 8 4
, where
' !
on with
? R
+
has
R +
and let be the fan in ?G@ consisting of the cones generated by all proper subsets + + +
6 . This is the normal fan of the / -simplex of and by Example 2.3.14 and Exercise 2.3.6. (You + will check the details to verify that this gives the usual affine open cover of in Exercise 3.1.3).
To relate this example to the discussion in Chapter 2, note that we have differ+ + ent embeddings of according to which is used to construct the Veronese + Y mapping. However, is the normal fan of for all , and the varieties obtained this way for different are isomorphic.
Example 3.1.11. We classify all -dimensional normal toric varieties as follows. . The only cones are the intervals 6 R % 8 We may assume ? R and ? @ R ( 4FR - and the trivial cone H R . It follows that there are only four and 6 R
possible fans, which gives the following list of toric varieties:
H . which gives M
6 H and 6 ( H . both of( which give
6 6 ( H . (which gives
Here is a picture of the fan for
:
6 ( This is the fan of Example 3.1.10 when /
R .
6
§3.1. Fans and Normal Toric Varieties
(
111
+
4 (8
P
4 8
? with ? @ Y andY ( ? Y@ Example 3.1.12. Let ? R ? . Let Y ' be the fan in 4 ? ' 8 @ as in Example R 3.1.10. Then is a fan in ? , consisting of all cones 6 ( 6 , where+ 6 ' isP in Y ' . By Example 2.4.8, the corresponding toric variety is the product .
σ10
σ00
σ11
σ01
Figure 3. A fan
with
Y ! Z
BC B C B C B C
D.47A% 203 - 8 D.47A% 2 3 - 8 D.47A% 2 3 - 8 D.47A% 203 - 8
? @ pictured in Figure 3. as above. Then
When / R , R , we obtain the fan 6 Label the 2-dimensional cones 6 ' R 6 '
F% ' -
( F% ' " -
( ( F% ' " " ( F% ' " - 4 (Let S P 8 4 ? ( 8 @ be as above, but let Y be? the, with Y R fanYSconsisting GP Y of the cone?G@ ( together with all its faces. Then is a fan in + ( and the the cor 6 responding toric variety is . The case was studied in Example 3.0.14.
Examples 3.1.12 and 3.1.13 are special cases of the following general construction, whose proof will be left to the reader (Exercise 3.1.4). Proposition 3.1.14. Suppose we have fans is a fan in ?
(% ?
Y ( Y
and
6
Y (
R 6 ( 6
in
4? ( 8@
6 ' . Y '
- 6
6
6
and
Y
in
4 ? 8 @
. Then
Chapter 3. Normal Toric Varieties
112
(
Example 3.1.15. The two cones 6 and 6 in ? @ R from Example 3.1.4 (see Y Figure 1), together with their faces, form a fan . By comparing the descriptions ;3 it of the coordinate rings of ( what 6 given there with we did in Example 3.0.8, ! is easy to check that , where is the blowup of at the R 354 ' R ')+ (*'V8 (Exercise 3.1.5). origin, defined as
( +
and set R with standard basis Generalizing this, let ? R + Y fan in ? @ consisting of the cones generated by all .Let +, benotthecontaining +
( +, . Then the toric variety 6 is subsets of isomorphic to the blowup of at the origin (Exercise 3.0.8).
(
and consider the fan Y
.
in ? @ R
Example 3.1.16. Let consisting of the four cones 6 ' shown in Figure 4, together with all of their faces. The correσ4 (−1, r)
σ1
σ3
σ2
sponding toric variety
6
with
Figure 4. A fan
is covered by open affine subsets,
3 R BC D47A% ' - 8 4
3 R BC D47A% ' " ( - 8 4
4
3 R BC D47A% ' " ( * ' 6
and glued according to (3.1.3). We call
" (-8
3 R BC D47A% ' " ( * ' " 4
-8
the Hirzebruch surface
L
4 ( 8 R = I> 4FR ( R ( 8 . We have 6 7 R consider the cone 6 R = I> 4FR ( R 8 ! ; , so the situation is similar to the case we studied in Example 1.2.21. Indeed, there is a change of coordinates defined by a matrix in 4 JX8 that takes 6 to the cone with @ R from that example. It follows that 4 3 354 ' TR 8 ! (Exercise 3.1.6). This is the there is an isomorphism rational normal cone , hence has a singular point at the origin. The toric variety 6 is singular because of the singular point in this affine open subset. We also saw $ 4 .8 a polytope such that the toric variety Y%$ coincides with the fan shown above. in Example 2.4.6, and the normal fan
Y
6
There is a dictionary between properties of and properties of that generalizes Theorem 1.3.12 and Example 1.3.20. We begin with some terminology. The first two items parallel Definition 1.2.16. Definition 3.1.18. Let (a) (b)
Y We say
Y We say Y
(c) We say
Y ! ?@
be a fan.
is smooth (or regular) if every cone 6 in
Y
is smooth (regular).
Y is simplicial if every cone 6 in is simplicial. Y R 3 6 6 is all of ? @ is complete if its support
.
Chapter 3. Normal Toric Varieties
114
Theorem 3.1.19. Let
6
be the toric variety defined by a fan
Y ! ?A@
.
Y (a) is a smooth variety if and only if the fan is smooth. 6 is an orbifold (that is, 6 has only finite quotient singularities) if and only (b) Y if the fan is simplicial. 6 is compact in the classical topology if and only if Y is complete. (c) 6
Proof. Part (a) follows from the corresponding statement for affine toric varieties, Theorem 1.3.12, because smoothness is a local property (Definition 3.0.10). In part (b), Example 1.3.20 gives one implication. The other will be proved later in the book. A proof of part (c) will be given in §3.4.
The blowup of at the origin (Example 3.1.15) is not compact, since the . The Hirzebruch support of the cones in the corresponding fan is not all of < surfaces from Example 3.1.16 are smooth and compact because every cone in . The variety the corresponding fan is smooth, and the union of the cones is 4 .8 from Example 3.1.17 is compact but not smooth. It is an orbifold (it has only finite quotient singularities) since the corresponding fan is simplicial.
Exercises for §3.1.
[ satisfy the compatibility G 3.1.2. Let be a variety obtained by gluing affine open subsets /&G r 8 along open subsets Gr ( *Gr by isomorphisms r ( 6>G r ( G ( r . Show that is separated when the image r G ( defined by RKS')(M Ke2p!r ( RKS'n' is Zariski closed for all sZ/2 * . of 6 GDr ( h GD" 3.1.3. Verify that if is the fan given in Example 3.1.10, then . 3.1.4. Prove Proposition 3.1.14. be the standard 3.1.5. Let , let 27+6+6+72 basis and let ( 8 . Let be the set of cones generated by all subsets of / 26+7+6+92 8 not containing / 26+7+6+9 2 8 . (a) Show that is a fan in .
3.1.1. Let be a fan in . Show that the isomorphisms conditions from §0 for gluing the together to create .
(b) Construct the affine open subsets covering the corresponding toric variety , and give the gluing isomorphisms. (c) Show that is isomorphic to the blowup of at the origin, earlier in described Exercise 3.0.8. Hint: The blowup is the subvariety of given by + + . Cover by affine open subsets + and compare those affines with your answer to part (b).
% d c # d 7c O5
Ne'
uM
t
(
4
(
`2+' 4 (b) Show that =?7@1A H 9J ' % "d # c 4E'^f o . 3.1.7. In (.P4 , consider the fan with cones /E `8 , ! ?1 ' , and ! `?1p# ' . Show that .
3.1.6. In this exercise, you will verify the claims made in Example 3.1.17. ( (a) Show that there is a matrix defining a change of coordinates that takes the cone in this example to the cone from Example 1.2.21, and find the mapping takes to the dual cone.
§3.2. The Orbit-Cone Correspondence
115
§3.2. The Orbit-Cone Correspondence
5
6
In this section, we will study the orbits for the action of on the toric variety . Our main result will show that there is a bijective correspondence between cones Y 5 -orbits in 6 . The connection comes ultimately from looking at limit in and 5 defined in §1.1. points of one-parameter subgroups of A First Example. We introduce the key features of the correspondence between orbits and cones by looking at a concrete example.
for the fan Y from Figure 2 of §3.1. We have Example 3.2.1. as the set of points 4 8 , 5 R 47 M 8 ! Consider with homogeneous coordinates R P . For each R 4 * 8 . ? R , we have the corresponding curve in :
6
)
4 8 R 4 8
)
We are abusing notation slightly; strictly speaking, the one-parameter subgroup 47 8 47 8 is a curve in M , but we view it as a curve in via the inclusion M ! .
48
We start by analyzing the limit of as . The limit point in 4 * 8 . It is easy to check that the pattern on R is as follows: )
depends
limit is (1,1,0) ↓
limit is (1,0,0) limit is (0,1,0) ← limit is (1,0,1) limit is (1,1,1)
←
a
limit is (0,0,1)
↑
b
limit is (0,1,1) Figure 6.
*
for
For instance, suppose in . These points lie in the first quadrant. ! # 4 8 R 4 8 . Next suppose that * R Here, it is obvious that in , corresponding to points on the diagonal in the third quadrant. Note that
4 8 R 4 8#% 4 " 8
R *
since we are using homogeneous coordinates in . Then implies that # 4 " 8 R 4 8 . You will check the remaining cases in Exercise 3.2.1.
Chapter 3. Normal Toric Varieties
116
Y
The regions of ? described in Figure 6 correspond to cones of the fan . C D4 6 8 , In each case, the set of giving one of the limit points equals ? 9 C D4 6 8 is the relative interior of a cone 6 . Y . In other words, we have where Y recovered the structure of the fan by considering these limits!
5
Now we relate this to the -orbits in . By considering the description 47 SF 8 M , you will see in Exercise 3.2.1 that there are exactly seven -orbits in :
5
( R 4 ' *')(*'
R 4 ' *')(*'
R 4 ' *')(*'
R 4 ' *')(*'
R 4 ' *')(*'
R 4 ' *' ( *'
R 4 ' *')(*'
' ' R P for all 54 8 ' R and ' *' ( R P 54 8 ')( R and ' *' R P 54 8 ' R and ')(*' R P 54 8 ')( R ' R and ' R P R 4 8 8 ' R ' R and ' ( R P R 4 8 8 ' R ')( R and ' R P R 4 8.
8 8 8 8 8
This list shows that each orbit contains a unique limit point. Hence we obtain a correspondence between cones 6 and orbits by
#
48 .
. CX D4 6 8 We will soon see that these observations generalize to all toric varieties 6
corresponds to
)
for all
6
.
Points and Semigroup Homomorphisms. It will be convenient to use the intrinsic 4 3 description of the points of an affine toric variety given in Proposition 1.3.1. We recall how this works and make some additional observations. 4
23
Points of
3
are in bijective correspondence with semigroup homomorphisms 2+3 R BC D47A% 23 - 8 . 4 3 . Recall that 6 7 9-; and R
For each cone 6 we have a point of
4
3
defined by
,. 23 R
,. 2 3 9 6 I R 6 I 9 ; otherwise
This is a semigroup homomorphism since 6 7 9 6 I is a face of 6 7 . Thus, if , , Z . 203 and , , Z . 203 96 I , then , , Z . 23 96 I . We denote this 3 point by and call it the distinguished point corresponding to 6 .
3
The point is fixed under the (Corollary 1.3.3). If H
! 6
is a face, then
5
-action if and only if
!
6 R ! G ? @
. 3 . This follows since 6 I ! H I 4
.
Limits of One-Parameter Subgroups. In Example 3.2.1, the limit points of oneparameter subgroups are exactly the distinguished points for the cones in the fan of (Exercise 3.2.1). We now show that this is true for all affine toric varieties.
§3.2. The Orbit-Cone Correspondence Proposition 3.2.2. Let 6J! let . ? . Then
Proof. Given
#!
)
48
. CX . ?
be a strongly convex rational polyhedral cone and
#! D4 6 8 , then ! #
Moreover, if
?:@
117
. 6
, we have
exists in
4
3
) )
4 8 exists in 4 3 48 R 3 .
P 4 ) 4 8*8 203 exists in for all ,. ! P exists in for all ,. 2+3
#!
'
#
L W for all ,. 6 7 9 ; K, % 46 7 87 R 6 .
where the first equivalence is proved in Exercise 3.2.2 and the other equivalences are clear. This proves the first assertion of the proposition. In Exercise 3.2.2 you will also show that when . point corresponding to the semigroup homomorphism
! ) 4 8 2 6:3 9 ? , # defined by
is the
P ,. 6 7 9 ; R+ # 203 X C D 4 8 SG6 I (Exercise 1.2.2), and 6 , then K , L for all , . If . 0 2 3 I ,K L R if , . 96 . Hence the limit point is precisely the distinguished 3
point
.
Y
6
Using this proposition, we can recover the fan from cone by cone as in Example 3.2.1. This is also the key observation needed for the proof of Corollary 3.1.8 from the previous section.
354 ' R S8
Let us apply Proposition 3.2.2 to a familiar example.
R Example 3.2.3. Consider the affine toric variety studied in a number of examples from Chapter 1. For instance, in Example 1.1.17, we showed that is the normal toric variety corresponding to a cone 6 whose dual cone is = 6 7 R I> 4 ( ( B C D.47F% 6 7 9-; - 8 . and R (3.2.1)
In Example 1.1.17, we introduced the torus image of (3.2.2) Given (3.2.3)
%R 8
R 74 M 8
included in
4 ( 8 4 ( ( " ( 8 R 4 * 8 . ? R , we have the one-parameter subgroup ) 4 8 R 4 U " 8
as the
Chapter 3. Normal Toric Varieties
118
contained in , and proceed to examine limit points using Proposition 3.2.2. ! # ) 4 8 weexists * W and * W . These Clearly, in if and only if conditions determine the cone 6J! ?:@ given by
= 6 R I> 4 ( (
(3.2.4)
8
One easily checks that (3.2.1) is the dual of this cone (Exercise 3.2.3). Note also CX D4 8 * * , in which case the limit and that . 3 # ) 4 8 R 4 6 means 8 , which is the distinguished point .
The Torus Orbits. Now we turn to the Y has a distinguished point 3 cone 6 .
5 .
3
6
. We saw above that each
6 ! . This gives the torus orbit 3 6 !
-orbits in 4
46 8 R 5
4 8 In order to determine the structure of 6 , we need the following lemma, which you will prove in Exercise 3.2.4. Lemma 3.2.4. Let 6 be a strongly convex rational polyhedral cone in ?Q@ . Let ? 3 4 8 be the sublattice of ? spanned by the points in 6T9 ? , and let ? 6 R ? ? .
3
(a) There is a perfect pairing
9 ; ? 4 6 8 K L 6 I ? . induced by the dual pairing K L ;
(b) The pairing of part (a) induces a natural isomorphism
I.*4 6 I 9-; J M 8 5 3 5 3 R ? 4 6 8 * M is the torus associated to ? 4 6 8 . where
4 6 8 ! 4 3 , we recall how . 5 acts on semigroup homomorTo study 2+3 , then by Exercise 1.3.1, the point 4 3 phisms. If . is represented by ,
P 48 4 8 ,
is represented by the semigroup homomorphism
(3.2.5)
, R
'
Lemma 3.2.5. Let 6 be a strongly convex rational polyhedral cone in ?Q@ . Then
where ?
46 8
4 6 8 R 23 4 , 8 R P , . 6 I 9 ; I. )4 6 I 9-; J M 8 5 3 ,
.*
is the lattice defined in Lemma 3.2.4.
Z
23 4 , 8 R P 5
R
Proof. The set is invariant under the action of
by (3.2.5).
,. 6 I 9 ;
contains
3
and
Next observe that 6 I is the largest vector subspace of ; @ contained in 6N7 . 2 3 R
Z , then restricting 7 Hence 6 I 9 ; is a subgroup of . If . 6 9 ; 203 to , . 9 6 I R 6 I 9V; yields a group homomorphism 6 I 9V; M
§3.2. The Orbit-Cone Correspondence
119
I 9 ; M is a group homomorphism , we . Z by defining 4 8 , . 6 I 9-; 4 , 8 R , if
6 (Exercise 3.2.5). Conversely, if obtain a semigroup homomorphism
It follows that
Z
,
I..*4 6 I 9-; J M 8 .
otherwise.
Now consider the exact sequence (3.2.6) Tensoring with
M
R+ ? 3 R+ ? R+ ? 4 6 8XR
and using Lemma 3.2.4, we obtain a surjection
* M R * M 5 R ? 5 3 R ? 4 6 8
I..*4 6 I 9-; J M 8
,
5 3 , I..*)4 6 I 9; J M 8 Z 5 -action, so that 5 acts transtively on Z . Then 3 . are compatible with the 5
Z 3 R 4 6 8 , as desired. R implies that The bijections
Z
The Orbit-Cone Correspondence. Our next theorem is the major result of this section. Theorem 3.2.6 (Orbit-Cone Correspondence). Let Y fan in ? @ .
6
be the toric variety of the
(a) There is a bijective correspondence
Y Q 5 6 -orbits in 4 6 8 , I..*)4 6 I 6 T 9 ; J M 8
!
Let / R ? @ . For each cone 6 . Y , 4 6 8 R / R ! 6 4 3
cones 6
(b)
in
(c) The affine open subset
.
4
is the union of orbits
3 R
is a face of
3
4H 8
4 8 4 H 8 , and (d) H is a face of 6 if and only if 6 !
4H 8 R
4 8 3 6 is a face of
4 8 where H denotes the closure in both the classical and Zariski topologies. For instance, Example 3.2.1 tells us that for , there are three types of cones and torus orbits:
Chapter 3. Normal Toric Varieties
120
4 8
4 8
5
R ! The cone R , which
! 4 8 R R corresponds
! to the orbit R satisfies . This is a face of all the other cones Y in , and hence all the other orbits are contained in the closure of this one by part (d). According to part (c), since there are no cones properly contained in 4 R 4 8 47 M 8 . ,
The three one-dimensional cones H give the torus orbits of dimension 1. Each of these is isomorphic to M . The closure ( of one of these orbits is one of the 354 ' ' 8 ! , a copy of . Note that each H is contained in coordinate axes two maximal cones.
Y ' fan correspond 4 8 4 8 4 8 6 of inthethetorus to the three fixed points action on . There are two of these in
The three maximal cones
the closure of each of the 1-dimensional torus orbits.
5 -orbit in 6 . Since 6 is covered by the 6 and 4 3 9 4 3 R 4 3 3 , there is a !
! 4 3 . We claim that R 4 6 8 . Note that
and consider those ,. 2 3 satisfying 4 , 8 R P . To prove the claim, let . In Exercise 3.2.6, you will show that these , ’s lie on a face of 6 7 . But faces of 687 are all of the form 6 7 9 H0I for some face H of 6 by Proposition 1.2.10. In other words, there is a face H of 6 such that
, . 23 4 , 8 R P R 6 7 9 H I 9 ; 4 This easily implies . (Exercise 3.2.6), and then H R 6 by the minimality of 0 2 3 4 8 R 6 I 9 ; , and then . 4 6 8 by Lemma 3.2.5. R , . , 6 . Hence
P
4 6 8 since two orbits are either equal or disjoint. This implies R
Proof of Theorem 3.2.6. Let be a 5 -invariant affine open subsets 4 3 Y with unique minimal cone 6 . part (a) follows immediately.
Part (b) follows from Lemma 3.2.5 and (3.2.6).
3
Next consider part (c). We know that is a union of orbits. If H is a face of 6 , 4 8 4 4 3 4 8 4 3 H H ! ! then implies that is an orbit contained in . Furthermore, 4 3 the analysis of part (a) easily implies that any orbit contained in must equal 4 H 8 for some face H of 6 .
4
4 8
H in the classical We now turn to part (d). We begin with the closure of 5 4 8 H topology, which we denote . This is invariant under (Exercise 3.2.6) and 4 6 8 ! 4 H 8 . Then hence is a union of orbits. Suppose that have 4 H 8 ! 4 3 , since otherwise 4 H 8 9 4 3 R , which would imply we 4 H 8 must 4 3 R 4 3 4 H 8 ! 4 3 , 9 it follows since is open in the classical topology. Once we have that H is a face of 6 by part (c). Conversely, let H be a face of 6 . To prove that 4 6 8 ! 4 H 8 , it suffices to show that 4 H 8 9 4 6 8 R P . We will do this by using limits of one-parameter subgroups as in Proposition 3.2.2.
Let point of
4
homomorphism corresponding to the distinguished ,besothe 4 semigroup , 8 R if , . H+I 9; , and otherwise. Let . CX D4 6 8 ,
§3.2. The Orbit-Cone Correspondence
121
4 8 . This semigroup homomorphism is P P 4 8 , , R ' 4 ) 4 8*8 4 , 8 R 4 8 . 4 H 8 for all . M since the orbit of is 4 H 8 . Now let . Note that CX D4 6 8 , L Since . , K if , . 6 7 SG6 4 I 3, and R if , . 6 I . It 4 8 ! 4 8 R #8 follows that exists as a point in by Proposition 3.2.2, and
4 8 4 represents a point in 6 . But it is also in the closure of H by construction, so
4 6 8 9 4 H 8 R P . This establishes the first assertion of (d), and
4H 8 R
4 8 3 6 and for each
. M
consider
48 R
)
is a face of
follows immediately for the classical topology.
It remains to show that this set is also the Zariski closure. If we intersect 4 3 with an affine open subset , parts (c) and (d) imply that
4H 8 9 3 R 3 4
3
4H 8
46 8
354768 ! 4 3 for the ideal 6 R K ' ,. H I 9 4 6 Z 8 7 9-; L ! F%04 6 Z 8 7 9; - R 2 3 (3.2.7)
4 8 6 This easily implies that the classical closure H is a subvariety of and hence
4 8 is the Zariski closure of H .
4H 8 Orbit Closures as Toric Varieties. In the example of , the orbit closures is a face of
containing
In Exercise 3.2.6, you will P show that this is the subvariety
also have the structure of toric varieties. The same is true in general. The torus that 85 from Lemma 3.2.5. The corresponding fan can be acts is the quotient torus Y described as follows. Consider the set of cones 6 in containing H as a face. For 4 8 each such 6 , let 6 be the image cone in ? H @ under the quotient map
? @ R ? 4H 8@ R
obtained from (3.2.6). Then (3.2.8) is a fan in ?
4H 8@
B DB 4H 8 R 6 ! ? 4H 8@ H
(Exercise 3.2.7).
Proposition 3.2.7. For each cone . the toric variety
H
in
Y
is a face of 6
, the orbit closure
4H 8
is isomorphic to
Proof. This follows from parts (a) and (d) of Theorem 3.2.6 (Exercise 3.2.7).
Y
Example 3.2.8. Consider the fan in ?A@ R shown in Figure 7 on the next Y Y page. The support of is the cone in Figure 2 of Chapter 1, and is obtained from
this cone by adding a new -dimensional cone H in the center and subdividing. The 4 8 H has dimension by Theorem 3.2.6. By Proposition 3.2.7, the orbit orbit 4 8 Y closure H is constructed from the cones of containing H and then collapsing
Chapter 3. Normal Toric Varieties
122
z
y τ
x
Figure 7. The fan
H
to a point in 4 8 so that H
and its -dimensional cone
? 4( H 8 @ R ( 4 ? ? 8 @ .
. This clearly gives the fan for
( (,
Final Comments. The technique of using limit points of one-parameter subgroups to study a group action is also a major tool in Geometric Invariant Theory as in [71], where the main problem is to construct varieties (or possibly more general objects) representing orbit spaces for the actions of algebraic groups on varieties. We will apply ideas from group actions and orbit spaces to the study of toric varieties in Chapters 5 and 14. We also note the observation made in part (d) of Theorem 3.2.6 that torus orbits have the same closure in the classical and Zariski topologies. For arbitrary subsets of a variety, these closures may differ. A torus orbit is an example of a constructible subset, and we will see in §3.4 that constructible subsets have the same classical and Zariski closures since we are working over . Exercises for §3.2. 3.2.1. In this exercise, you will verify the claims made in Example 3.2.1 and the following discussion. are as claimed in the (a) Show that the remaining limits of one-parameter subgroups example.
4
(b) Show that the
F 'n4 orbits in 4 are as claimed in the example.
(c) Show that the limit point equals the distinguished point in each case.
}{ |
3.2.2. Let consider
, !n' , where ,'6m h ~
of the corresponding cone
be a strongly convex rational polyhedral cone. This exercise will is an arbitrary function.
§3.2. The Orbit-Cone Correspondence
}{ |
, An'
123
(a) Prove that exists in if and only if . Hint: Consider a finite set of characters
{}|
, !n'
}{ |
{}|
F, An'n' exists in ( , .
such that
for all
exists in , prove that the limit is given by the semigroup homo(b) When morphism that maps to .
F, !n' '
3.2.3. Consider the situation of Example 3.2.3. (a) Show that the cones in (3.2.1) and (3.2.4) are dual. (b) Identify the limits of all one-parameter subgroups in this example, and describe the Orbit-Cone Correspondence in this case.
# uC( # defines an automorphism of o and the corresponding linear map on
(c) Show that the matrix
the cone
to .
(d) Deduce that the affine toric varieties tion 1.3.15.
maps
and are isomorphic. Hint: Use Proposi-
3.2.4. Prove Lemma 3.2.4. 3.2.5. Let be as defined in the proof of Lemma 3.2.5. In this exercise, you will complete the proof that is a -orbit in .
(a) Show that if , then is a group homomorphism. (b) Deduce that
~
6
h
has the structure of a group.
(c) Verify carefully that we have an isomorphism of groups
m|
2 '.
6 h be a semigroup homomorphism giving a point of . Prove that / O k 'U(*T `8U( for some face of . (b) Show ' is invariant under the action of ~ . (c) Prove that ' is the variety of the ideal defined in (3.2.7). 3.2.7. Let be a cone in a fan , and let < E ' be as defined in (3.2.8). (a) Show that < & ' is a fan in . ' . (b) Prove Proposition 3.2.7. ~ on the affine toric variety . Use parts (c) and (d) of 3.2.8. Consider the action of ~ acting on . Theorem 3.2.6 to show that 5' is the unique closed orbit of 3.2.9. In Proposition 1.3.16, we saw that if is a face of the strongly convex rational polyhedral cone in then ( =?7@mFIH J!' is an affine open subset of ( =?7@1AIH J!' . In this exercise, we will show that the converse is also true, i.e., that if % and the induced map of affine toric varieties ] 6 h is an open immersion, then is a face of . (a) Let P2n Q , and assume t Q . Show that }{}| !n' }{}| !n' W+ 3.2.6. This exercise is concerned with the proof of Theorem 3.2.6. (a) Let
Chapter 3. Normal Toric Varieties
124
}{ |
!n'
An'
{}|
(b) Show that and
of points as semigroup homomorphisms. (c) Deduce that
P2n S
are each in . Hint: Use the description
, so is a face of .
3.2.10. In this exercise, you will use Proposition 3.2.2 and Theorem 3.2.6 to deduce Corollary 3.1.8 from Theorem 3.1.7. (a) By Theorem 3.1.7, and the results of Chapter 1, a separated toric variety has an open cover consisting of affine toric varieties + for some collection of cones + . is also affine. Hint: Use the fact that is separated. Show that for all , +
(
( 4
2
(
+ . (b) Show that + is the affine toric variety corresponding to the cone Hint: Exercise 3.2.2 will be useful. ,+% , then show that is a face of both + and . Hint: Use Exercise 3.2.9. (c) If
(d) Deduce that
+ and all their faces.
for the fan consisting of the
§3.3. Equivariant Maps of Toric Varieties
Recall from §3.0 that if and are varieties with affine open covers R 4 Z is a Zariski-continuous and R , then a morphism of varieties mapping such that the restrictions (
4
9
" 4 Z 8 R 4
are regular in the sense of Definition 3.0.1 for all
When
and
"M
4
Z
4
.
are normal toric varieties, the results on mappings of affine toric varieties from Propositions 1.3.14 and 1.3.15 yield a class of morphisms whose construction comes directly from the combinatorics of the associated fans. The goal of this section is to study these special morphisms.
(
Y(
4 (8
Y
Definition 3.3.1. Let ? ? be two lattices a fan in ? @ and YS( a fan 4 8 @ . A -linear mapping ( ? with ? in ? is compatible with the fans Y if for every cone 6 ( in Y ( , there exists a cone 6 in Y such that 4 ( 8 and . @ 6 !6
(
(
Y
R Example 3.3.2. Let ? basis and let be the fan from Figure 4 < with 6 R in §3.1. By Example 3.1.16, ( is the Hirzebruch surface . Also let ? Y and consider the fan giving :
6 (
6
as in Example 3.1.11. The mapping
? (%R ? * ( R+ * Y and Y since each cone of Y is compatible with the fans
Y
. If
RP
, on the other hand, the mapping
? ( R+ ?
*( Y is not compatible with these fans since 6 .
R
maps onto a cone of
Y
does not map into a cone of .
§3.3. Equivariant Maps of Toric Varieties
125
Toric Morphisms. For our purposes, it will be convenient to take the result of part (a) of Proposition 1.3.14 as the definition of a toric morphism in this context. The discussion from §1.3 shows that this is consistent with what we did for affine toric varieties.
* ,
6
YS(
4 (8
@ Definition 3.3.3. Let , be normal toric varieties, with a fan in ? Y 4 8 6
6 and in ? is toric if maps the torus 5 ! a 6 faninto 5 @ ! . A 6morphism and is a group homomorphism.
6
The proof of part (b) of Proposition 1.3.14 generalizes easily to show that any 6 6 is an equivariant mapping for the 5 - and toric morphism 5 -actions. That is, we have a commutative diagram
5 0 6
(3.3.1)
where
(
and
/
*-,
6
/
6 5
6
give the torus actions.
( ?
Our first result in this section shows that toric morphisms are in bijective correspondence with -linear mappings ? Y( Y compatible with the fans and . Theorem 3.3.4. Let ?
(Q
( ?
6
6
that are
4 ? ' 8 @ , R . Y ( and Y , then -linear map that is compatible with Y '
be lattices, and let
(a) If ? ? is a there is a toric morphism
be a fan in
such that * , is the map ? (/ M R ? M is a toric morphism, then induces a -linear (b) Conversely, if ? ( ? that is compatible with the fans Y ( and Y . map ( Y ( YS( and Proof. To prove part (a), let 6 be a cone in . Since is compatible with 4 (8 Y Y , there is a cone 6 . with @ 6 3 3 ! 6 3 . Then Proposition 1.3.15 shows that induces a toric morphism . Using the general criterion for 3 gluing morphisms from Exercise 3.3.1, you will show in Exercise 3.3.2 that the . Moreover, is toric because glue together to give a morphism 5 5 , which is easily seen to be the group ( taking 6 R gives ? ( ? . homomorphism induced by the -linear map we show first that the toric morphism induces a -linear map ? For( part? (b), */, . This follows since is a group homomorphism. Hence, given . ? ( , the one-parameter subgroup M 5 can be composed with * , */, M 5 . This defines an to give the one-parameter subgroup 4 8 . ? . It is straightforward to show that ? ( ? is -linear. element 6
6
+*
6
+*
6
4
4
6
)
6
)
Chapter 3. Normal Toric Varieties
126
YF(
Y
It remains to show that is compatible with the fans 5 5 -orbit ( ! 6 is mappedandinto a. Because of the equivariance (3.3.1), each 5 -orbit
6 Orbit-Cone Correspondence (Theorem 3.2.6), each ( R! 4 6 (. 8 Byforthesome ( Y ( 4 6 is8 5 -orbit is R -orbit cone 6 in , and similarly each Y . Furthermore, if H ( ! 6 ( is a face, then by the same for some cone 6 in Y 4 4 H ( 8*8 ! 4 H 8 . such that reasoning, there is some cone H in
H
in this situation must be a face of 6 . This follows since
4 6 We
4 H 8 that 8 ! claim by part (d) of Theorem 3.2.6. Since is continuous in the Zariski 4 ( 8 4 8
H H . But the only orbits contained in the closure of ! topology, 4 H 8 are the orbits corresponding to cones which have H as a face. So H is a face of 6 . It follows from part (c) of Theorem 3.2.6 that also maps the affine open 3 4 3 ! 6 into 4 ! 6 , i.e., subset
4 38 ! 3 3 3 It follows that induces a toric morphism , which by Proposition 1.3.15 4 ( 8 ! 6 . Hence is compatible with the fans YS( and Y . implies that @ 6 4
(3.3.2)
4
4
4
First Examples. Here are some examples of toric morphisms defined by mappings compatible with the corresponding fans. Example 3.3.5. Let ?
( R
and ?
R
, and let
? ( R+ ? * ( R * is compatible with the fans be the first mapping in Example 3.3.2. We saw Y of the Hirzebruch surface < and Y of ( . that 3.3.4 implies that there < ( ( Theorem is a corresponding toric morphism . We will see ( later in this section < that this mapping gives the structure of a -bundle over . + , the multiplicaY Example 3.3.6. Let ? R and be a fan in ? @ . For .
tion map
Y is compatible with 6 6
? R+ ?
* R+
*
. By Theorem 3.3.4, there is a corresponding toric morphism 5 ! 6 is the group endomorphism whose restriction to
! *-, 4 ( +98 R 4 ( + 8
. R For a concrete example, let be the fan in ?A @ from Figure 2 and take R Then we obtain the morphism defined in homogeneous coordinates 4 ' *')(*' 8 R 4 ' *' ( *' 8 . by
Y
Sublattices of Finite Index. We get an interesting toric morphism when a lattice Y ? Z has finite index Zin a larger lattice ? . If Y is a fan in ? @ , then we can view Z as a fan either in ? @ or in ? @ , and the inclusion ? with the ? is compatible Y Z 6 5 6 5 fan in ? @ and ? @ . As in Chapter 1, we obtain toric varieties and
§3.3. Equivariant Maps of Toric Varieties
127
depending on which lattice we consider, and the inclusion ? morphism
5 R 5
Z
induced by the inclusion
Z
a sublattice of finite index in ? Z .beThen
and let
5 +R 5
? Z ?
induces a toric
6
6
Proposition 3.3.7. Let ? ? @ R ? @ Z . Let R ? ?
Z ?
Y
be a fan in
6
6
presents
6
5
as the quotient
6
5
.
Proof. Since ? has finite index in ? , Proposition 1.3.18 shows that the finite R ? ? Z is the kernel of 5 5 . It follows that acts on 6 5 . group 5 4 34 5 This action is compatible with the inclusion ! 4 6 34 5 for each cone 6 . Y . 4 34 5 Using Proposition 1.3.18 again, we see that , which easily implies 6 5 6 5 that .
We will revisit Proposition 3.3.7 in Chapter 5, where we will show that the 6 5 6 5 map is a good geometric quotient.
Y
5
Example 3.3.8. Let ? R , and be the fan shown in Figure 5, so is 4
. 8 Z . Let ? be the sublattice of isomorphic to the weighted projective space
? Z given by ? Z R 4 * ( 8 . ( ? I , so ? Z has index 2 in ? . Note that ? is generated by R , R and that
Z
R R ( R5 R R ( R
6
. ? Z
Let ? ? Z be6 the map. It is not difficult to see that with respect 5 inclusion to the lattice ? , (Exercise By Theorem 3.3.4, the -linear 3.3.3). 4
.8 , and by Proposition 3.3.7, it map induces a toric morphism 4
. 8 . Z follows that for R ? ?
σ2
Figure 8. The semigroups
and
The cone 6 from Figure 5 has the dual cone 6/7 shown in Figure 8. It is Z dual to ? Z . One checks instructive to consider how 6 7 interacts with the lattice ; Z 4 * .8 * . and 6 7 R = I> 4 (XR 4R 8 . In Figure 8, the that ;
Chapter 3. Normal Toric Varieties
128
Z
points in 67 9 ; are shown in white, and the points in 6/7 9 ; not in 6N7 9 ; are shown in black. Note that the picture in 6 7 9 ; is the same (up to a change 4 JX8 Figure 10 from Chapter 1. This shows again that of coordinates in 4 .8 contains the affine) as 4 3 5 open subset isomorphic to the rational normal 4 3 5 is smooth. The other affine open subsets cone . On the other hand ( in both 4 .8 . corresponding to 6 and 6 are isomorphic to and in
6
Torus Factors. A toric variety has a torus factor if it is equivariantly isomorphic to the product of a nontrivial torus and a toric variety of smaller dimension.
Proposition 3.3.9. Let equivalent: (a)
6
6
Y
be the toric variety of the fan . Then the following are
has a torus factor.
(b) There is a nonconstant morphism
M .
6
. YS4 8 , do not span ?G@ . 6 6 47 M 8 6 Proof. If for and some toric variety , then a non6 7 4 8 7 4 M8 M. trivial character of M gives a nonconstant morphism 6 M is a nonconstant If morphism, then Exercise 3.3.4 implies that 5 is ' P where . M , . ; . Multiplying by the( restriction of to P " , we may assume that *-, R ' . Then is aand toric morphism coming from M ? a surjective homomorphism . Since comes from the trivial fan, Y Y S 4
8 maps all cones of to the origin. Hence . for all . , so that the do not span ? @ . Y 4 8 S Finally, suppose that the , . span a proper subspace of ?:@ . Then B + C B Z ) 4 Y S 4
* 8 8 Z is . is proper sublattice of ? such that ? ? ? R 9 ? Y Z Z Z Z Z Z ? Y . Furthermore, with ? R ? can torsion-free, so ? has a complement ? Y Z Z Y Y Z Y Z Z R be regarded as a fan in ? @ , and then is the product fan , where Y ZZ ZZ is the trivial fan in ? @ . Then Proposition 3.1.14 gives an isomorphism 6 6 5 5 6 5 5
47 M 8 + "
!
? @ R / and ! ? @ Z R . where (c) The
,
In later chapters, torus varieties without torus factors will play an important role. Hence we state the following corollary of Proposition 3.3.9. Corollary 3.3.10. Let equivalent: (a)
6
6
Y
be the toric variety of the fan . Then the following are
has no torus factors.
6 M is constant, i.e., 4 . YS4 8 , span ? @ .
(b) Every morphism (c) The
,
6
" 8M R M.
We can also think about this from the point of view of sublattices.
§3.3. Equivariant Maps of Toric Varieties
129
? @ R / , ! ? @ Z R G @ .
Z
! ? be a sublattice with Proposition 3.3.11. Let ? Y Z Let be a fan in ? @ , which we can regard as a fan in ? (a) If ?
Z
.
is spanned by a subset of a basis of ? , then we have+ an isomorphism
5 5 5 5 5 5 47 M 8 "
6
6
(b) In general, a basis for ? 6 of finite index. Then
6
5
Z
5 6
6
? ZZ ! ?
can be extended to a basis of a sublattice is isomorphic to the quotient of+
5
5 5
ZZ by the finite abelian group ? ? .
5 547 M 8 "
6
Proof. Part (a) follows from the proof of Proposition 3.3.9, and part (b) follows from part (a) and Proposition 3.3.7.
Y Z
Y
Y
Refinements of Fans and Blowups. Given in ? @ , a fan Y Z R a Y fan Y Z Y Y refines if every cone of is contained in and . Hence every cone of is a union YZ YZ refines Y , the identity mapping on ? is automatically of cones of . When Y Z Y 6 6 . compatible with and . This yields a toric morphism
Y Z
Example 3.3.12. Consider the fan in ? in Figure 1 from §3.1. = pictured Y > I 4 ( 8 and This is a refinement of the fan consisting of its faces. The 6 6 R 354 ' RQ' ( 'V8 ! are and corresponding toric varieties ( at the origin (see Example 3.1.15). The identity map , the blowup of on( ? induces a toric morphism . This “blowdown” ( morphism maps ! to . and is injective outside of
in .
+
We can generalize this example and Example 3.1.5 as follows.
Y
= I> 4 ( + 8
Definition 3.3.13. Let be a fan in ?@ . Let 6 R Y ( + smooth cone in , so that is a basis for ? . Let R Y Z46 8 be the set of all cones generated by subsets of ( + . Then
( +
+
be a and let not containing
Y M 4 6 8 R 4 Y S 6 8 Y Z 4 6 8 Y is a fan in ? @ called the star subdivision of along 6 . = I> 4 ( 8 ! ?@ be a smooth cone. Example 3.3.14. Let 6 R Figure 9 on the next page shows the star subdivision of 6 into three cones = I> 4 ( 8 = I> 4 ( 8 = I> 4 8 Y 4 8 The fan M 6 consists of these cones, together with their faces. Y 4 8 Y Proposition 3.3.15. M 6 is a refinement of , and the induced toric morphism 6 3 R+ 6 3 6 -3 6 makes the blowup of at the distinguished point corresponding to the cone 6 .
Chapter 3. Normal Toric Varieties
130
u0
u2 u1
u3
Figure 9. The star subdivision
Y 4 8
Y
Proof. Since and M 6 are the same outside the cone 6 , without loss of generY + fan consisting of 6 and all of its faces, ality, we may reduce to the case that is the
6 4 3 and is the affine toric variety .
Under the Orbit-Cone Correspondence (Theorem 3.2.6), 6 corresponds to the 3 distinguished point , the origin (the unique fixed point of the torus action). By Theorem 3.3.4, the identity map on ? induces a toric morphism
3-
6
4
3 +
3
+ It is easy to check that the affine open sets covering are the same as for the blowup of at the origin from Exercise 3.0.8, and they are glued together in the same way by Exercise 3.1.5.
3
4 8 . In this notation, the
+ +8 The blowup at is sometimes denoted 7 4 blowup of at the origin is written . 6
6
6
The point blown up in Proposition 3.3.15 is a fixed point of the torus action. In some cases, torus-invariant subvarieties of larger dimension have equally nice+ ( + of blowups. We begin with the affine case. + The standard basis 3 = = > I 4 ( + 8 4 > I 4 ( 8 , R gives 6 R , and the face H R with / , gives the orbit closure
4 H 8 R A + "
4 8 R ( and consider the fan To construct the blowup of H , let Y M 4 H 8 R = I> 4 8 ! + . ( ! P . (3.3.3)
= I> 4 ( 8 = ( ! ? @ and H R I> 4 8 . 6 into the cones = >I 4 ( 8 = I> 4 8
Example 3.3.16. Let 6 R H The star subdivision relative to subdivides
§3.3. Equivariant Maps of Toric Varieties
as shown in Figure 10. The fan their faces.
Y M 4H 8
131
consists of these two cones, together with
e0
e2
τ e1
e3
Figure 10. The star subdivision
+" + 6 For the fan (3.3.3), the toric variety is the blowup+ of . Y 4 8 +" ! R To see why, observe that M H is a product fan. Namely, , and Y M 4 H 8 R Y ( Y Y ( .47 8 (coming from = I> 4 ( 8 ) and Y is the where + is the fan for " (coming from = I> 4 U ( + 8 ). It follows that fan for 6 R 47 8X + " " ( .47 + 8 is built by replacing . with Since follows ( +6 " R .47 8 " is built by replacing Q + " ! , + itwith " that . the 47 + 8 separates directions through the origin in , while +" intuitive idea is that 8 6 47 R the blowup normal directions to + . One can also study 47separates + 8 by working in on affine pieces given by Y 4 8 M H the maximal cones of —see [76, Prop. 1.26].
We generalize (3.3.3) as follows.
Y
+
Definition 3.3.17. Let be a fan in ?@ and assume H . Y H R & that all cones of containing are smooth. Let Y containing H , set cone 6 .
the property - ( has and for each
Y 3M 4 H 8 R = I> 4 8 ! 6 4 8 H 4 8 ! P Y Then the star subdivision of relative to H is the fan Y M 4 H 8 R 6 . Y H !P 6 Y 3 4 8 3 M H
Y
.
Chapter 3. Normal Toric Varieties
132
The fan
Y M 4H 8
4H 8.
Under the map , ; 4 8 closure H R
Y
is a refinement of 6
and hence induces a toric morphism
6
6
becomes the blowup
4 8
6
of
6
along the orbit
In Chapters 10 and 11 we will use toric morphisms coming from refinements of fans to resolve the singularities of toric varieties. Exact Sequences and Fibrations. Next, we consider some of the local structure of toric morphisms. To begin, consider a surjective -linear mapping
Z ? ? Y Y Z Z If in ? @ and in ? @ are compatible with , then we have a corresponding toric morphism R 4 8 , so that we have an exact sequence Now let ? R ? RR ? Z R (3.3.4) R+ ?
6
It is easy to check that
6
R 6 . Y 6 ! 4 ? 8 @ Y 4 8 @V! ? @ . By Proposition 3.3.11, is a subfan of whose cones lie in ? (3.3.5)
Y
Z
6
5 6 5 5
? . Furthermore, is compatible with since ? ? Z
in ? @ . This gives the toric morphism
"
,
" 4 5 8 R
6
Y
in ?@ and the trivial fan
5 5
In fact, by the reasoning to ( prove Proposition 3.3.4,
5 6 5 5 6 lying over 5 ! 6 is identified with the product In other words, the part of 5 6 5 . We say this subset of 6 is a fiber bundle of and the toric variety 5 with fiber 6 5 . over Y When the fan has a suitable structure relative to , we can make a similar 6 statement for every torus-invariant affine open subset of . Y Y Z Y Definition 3.3.18. In the situation of (3.3.4), we say is split by and if there Y Y exists a subfan ! such that: Y bijectively to a cone 6 Z . Y Z such that 6 6 Z defines (a) maps each cone 6 . Y Y Z
6
(3.3.6)
a bijection
(b)
. Y YGiven cones 6 arises this way.
.
and 6
. Y , the sum 6
6
Y
lies in , and every cone of
§3.3. Equivariant Maps of Toric Varieties
Y
Theorem 3.3.19. If is split by 6 locally trival fiber bundle over 4 open affine subsets satisfying
Y Z
133
Y
6
and as in Definition 3.3.18, then 6 5 . That is, 6 has a coverisbya with fiber
"(4 8 4
6
5
4
YS4 6 Z 8 R 6 . Y 4 6 8 ! 6 Z . Then " ( 4 4 3 8 R 6 3 6 3 6 5 .4 3 YS4 Z 8 Y 9 YS4 6 Z 8 It remains to show that . Since 6 is split by Y YS4 6 Z8 , we may assume 6 R 4 3 . In other words, we are reduced to the and 9 Y Z Z case when consists of 6 and its proper faces. ? Z ? splits the exact sequence (3.3.4) provided $ A -linear map $ Z is the identity on ? . A splitting induces an isomorphism Z ? ? ? Y such that maps 6 bijectively to 6 Z . By Definition 3.3.18, there is a cone 6 . Z Using 6 , one can find a splitting $ with the property that $ @ maps H to H for all Y H . (Exercise 3.3.5). Using Definition 3.3.18 again, we conclude that 4 ? 8 @ ? Z ?@ @ 4 Y 4 ? 8 @ 8 4 Y Z ? Z 8 4Y 8 carries the product fan @ to the fan ?G@ . By ProposiProof. Fix 6
Z
in
Y Z
and let
tion 3.1.14, we conclude that
6 6 5 - 6
6 5 4
3
and the theorem is proved. Example 3.3.20. To complete the discussion from Examples 3.3.2 and 3.3.5, con< ( sider the toric morphism induced by the mapping
Y
F4 R (
6 . Y @ 4 6 8 R
8 . = I> 94 ( 8 Y Z These cones are mapped bijectively to the cones in under @ consists of the cones = >I 4 8 . = >I F4 R 8
The fans
Y
and
Y
are shown in Figure 11.
Y
. and 6 . < As we vary over all 6 < . Hence Theorem 3.3.19 shows that with fibers isomorphic to
6 5
Y
, the sums 6
Y
. Note also that
give all cones of ( is a locally trivial fibration over ,
(
6
Chapter 3. Normal Toric Varieties
134
Σ0 ↓
〉
Σ ↓
σ4 (−1, r)
σ1
σ3
〉
←Σ σ2
↓
↓
↓
Figure 11. The Splitting of the Fan
4 8
R where ? gives the vertical axis in Figure 11. ( This( fibration is not < globally trivial when , i.e., it is not true that . There( is some " 44 3 8 “twisting” on the fibers involved when we try to glue together the ( < 4 3 to obtain .
We will give another, more precise, description of these fiber bundles and the “twisting” mentioned above using the language of sheaves in Chapter 6.
4 8
Images of Distinguished Points. Each orbit contains 6 4 in8 a toric variety 3 a distinguished point , and each orbit closure 6 is a toric variety in its own right. These structures are compatible with toric morphisms as follows.
6
6
6
Lemma 3.3.21. Let be the toric morphism coming from a map Z Y Y Z Y , let 6 Z . Y Z be the and . Given 6 . ? ? thatY Z is compatible with 4 8 minimal cone of containing @ 6 . Then:
4 3 8 R 3 , where 3 . 4 4 6 8*8 ! 4 6 Z8 and (b) 3 (c) The induced map
46 8
3 . 4 6 Z08
46 8 ! 46 Z8. 4 6 8 4 6 Z 8 is a toric morphism. Z( Z Y Z contain 4 6 8 , then so does their intersecProof. First observe that if 6 6 . @4 8 Y Z tion. Hence has a minimal cone containing @ 6 . (a)
and
are the distinguished points.
§3.3. Equivariant Maps of Toric Varieties
135
4 8 . C D4 6 Z 8 4 3 8 R ! # 4 8 R ! # 4 4 *8 8 R ! # 4 8 R 3
To prove part (a), pick Z the minimality of 6 . Then )
. C D 4 6 8
and observe that )
by
)
where the first and last equalities use Proposition 3.2.2. The first assertion of part (b) follows immediately from part (a) by the equivariance, and the second assertion follows by continuity (as usual, we get the same closure in the classical and Zariski topologies).
3- 4 8 4 Z08
For (c), observe that 6 6 is a morphism that is also a group homomorphism—this follows easily from equivariance. Since the orbit closures 3 4 6 8X 4 6 Z 8 is a toric are toric varieties by Proposition 3.2.7, the map morphism according to Definition 3.3.3.
Exercises for §3.3.
/ 8
3.3.1. Let be a variety with an affine open cover + , and let be a second variety. is + be a collection of morphisms. We say that a morphism Let + + for all . Show that there exists such a obtained by gluing the + if if and only if for every pair ,
] 6 h
] 6 h ] ] O 4 (N] ]6 h 2 ],+ O 4 *( ] O 4 + 3.3.2. Let 2 be lattices, and let in ' , in ' be fans. Let ] 6! h 4 4 4 4 be a -linear mapping that is compatible with the corresponding fans. Using Exercise 3.3.1 above, show that the induced toric morphisms ] 6 h glue together to form a morphism ] 6 h . 3.3.3. This exercise asks you to verify some of the claims made in Example 3.3.8. 4. (a) Verify that with respect to the lattice , Q[
(b) Verify carefully that the affine open subset 6[ g , the rational normal cone of 4 degree 2. , gives a morphism ~ h . Here you will determine all 3.3.4. A character , " morphisms ~ h . (a) Explain why morphisms ~ h correspond to invertible elements in the coordinate ring of ~ . r (b) Let t and s . Prove that is invertible in IH 27+6+6+72n J and that all invertible elements of IH 26+6+7+62 J are of this form. ~ are of the form for (c) Use part (a) to show that all morphisms ~ 0h on and .
3.3.5. Let and let and be cones in ] 6 h be a surjective -linear mapping
and respectively with the property that ] maps
bijectively onto . Prove that ] has a splitting 6! Dh such that maps to . 3.3.6. Let be the fan obtained from the fan for 4 in Example 3.1.9 by the following process. Subdivide the cone into two new cones and 4 4 4 4 by inserting an edge `?mp # 4 '.
Chapter 3. Normal Toric Varieties
136
explicitly and give the gluing homomorphisms. (b) Show that the resulting toric variety is smooth. (c) Show that is isomorphic to the Hirzebruch surface . (a) Construct an affine open cover for
nm2612 1'
3.3.7. Let be the fan obtained from the fan for ( in Example 3.1.17 by the following process. Subdivide the cone into two new cones and by inserting an edge . (a) Construct an affine open cover for explicitly and give the gluing homomorphisms.
`?1n# 4 '
4
(c) Construct a morphism of .
434
is smooth.
]6 h
(b) Show that the resulting toric variety
4
and determine the fiber over the singular point
(d) One of our smooth examples is isomorphic to
. Which one is it?
( / W2 o&'UO I( m8$MF 'p4 4 4 and (M/ W W26 1'ZO2' 2 - |Wz >8 < . Also let ) ( &4 + ) . (a) Let D( Prove that the map h F 'n4 defined by ` `2 6 m':jh 2 ' induces an exact ) ) sequence #Dh #Dh #h #h `+ `?1 2 4 'L ' ( ( 4 . The inclusion Qh induces a toric (b) Let ( morphism [ h [ . Prove that this is the quotient map 4 h 4
for the above action of on 4 .
on . We will ) 3.3.8. Consider the action of the group study the quotient and its resolution of singularities using toric morphisms.
4
(c) Find the Hilbert basis (i.e., the set of irreducible elements) of the semigroup Hint: The Hilbert basis has four elements.
(d) Use the Hilbert basis from part (c) to subdivide . This gives a fan with Prove that is smooth relative to and that the resulting toric morphism
Q[ h [ C(C 4
.
O :O( .
is a resolution of singularities. This is a special case of the theory to be developed in Chapter 10.
A 'p4:- 4
m 'B( d # 12 c # d o '
(e) The group gives the finite set with ideal . ) Read about the Gr¨obner fan in [18, Ch. 8,§4] and compute the Gr¨obner fan of . The answer will be identical to the fan described in part (d). This is no accident, as shown in the paper [54]. There is a lot of interesting mathematics going on here, including the McKay correspondence and the -Hilbert scheme. See also [69] for the higher dimensional case.
ko F `2 26#&'
3.3.9. Consider the fan in shown in Figure 12 on the next page. This fan has five one-dimensional cones with four “upward” ray generators and one “downward” generator . There are also nine two-dimensional cones. Figure 12 shows five of the two-dimensional cones; the remaining four are generated by the combining the downward generator with the four upward generators.
$12 26E'927F `2$126&'
c h . (b) Show that h is a locally trivial fiber bundle over with fiber nm27m2 (1' . Hint: Theorem 3.3.19 and pm23 `27E' Cp#m23 `27E' (A 2 `27#E' (V . See Example 3.1.17.
(a) Show that projection onto the -axis induces a toric morphism
§3.3. Equivariant Maps of Toric Varieties
137
z
y x
Figure 12. A fan
in
p126m2 m'
(c) Explain how you can see the splitting (in the sense of Definition 3.3.18) in Figure 12. Also explain why the figure makes it clear that the fiber is ( .
)4
3.3.10. Consider the fan in
with ray generators
( 4 2e ( 2e 4 ( 4 2 o (M# and two-dimensional cones `?1A 2 '92 `?1A 2n ' 2 ! ?1! 2 ' . 4 4 ato one point. (a) Draw a picture of and prove that is the blowup of h such that (b) Show the a toric morphism ] 6 map ! 4 jh induces ] As that ' for s * and ] A ' is a union of two copies of meeting at a F 1' gives point. Hint: Once you understand ] A 1 ' , show that the fan for ] . (c) To get a better picture of , consider the map 6F 'n4Uh o defined by F12 n'B(L F o 2 4 237 2 4 ' 2 n' + nF ' 4 'y o be the closure of the image. Prove that and Let ( o Chi to gives the toric morphism ] of that the restriction of the projection part (b). o U is defined by the equations (d) Let d 2 c 2 2 $ be coordinates on o . Prove that 'c $ # 4 (* 2 d # c 4 (C `2 %d $ # &c (V `+ FsP' for sLV , and explain how Also use these equations to describe the fibers ] this relates to part (b). Hint: The twisted cubic is relevant.
This is a semi-stable degeneration of toric varieties. See [51] for more details.
Chapter 3. Normal Toric Varieties
138
§3.4. Complete and Proper The Compactness Criterion. We begin by proving part (c) of Theorem 3.1.19. Theorem 3.4.1. The following are equivalent for a toric variety (a)
6
! #
is compact in the classical topology.
(b) The limit (c)
Y
)
48
exists in
is complete (that is,
Y
6
3 6 6 R ?@
R
. ?
for all
6
.
.
).
6
Proof. First observe that since is separated (Theorem 3.1.5), it is Hausdorff as a topological space (Theorem 3.0.17). In fact, since the classical topology on 4 3 6 is a metric topology, is compact if and only if every each affine open set 6 sequence of points in has a convergent subsequence.
(b), assume that is compact and fix . ? . Given a sequence ) 4 8 . 6 . By compactness, . For M (a)converging to , we get the sequence this sequence has a convergent subsequence. Passing to this subsequence, we can ! ) 4 8
6 6 R . assume that . Because the union of the affine 34 # 3 Y , we may assume . 4 . Nowis take ,. 6 7 9 ; . The open subsetsP for 6 . ' 4 3 character is a regular function on and hence is continuous in the classical topology. Thus 6
'
P 4 8 R
#
P 4 ) 4 8*8 R
'
#
! P
, the exponent must be nonnegative, i.e., K , L-W for all , . , . 687 , so that . 4 6N7 8 7 R 6 . Then 67 9 ; . This implies K , L!W for) all 4 8 exists in 4 3 and hence in 6 . # Proposition 3.2.2 implies that
Since
!
48
# To prove (b) (c), take . ? and consider the limit . This lies 4 3 . in some affine open , which implies 6 Y 9? by Proposition Y 3.2.2. Thus every lattice point of ?G@ is contained in a cone of . It follows that is complete.
)
?:@ . In the case / R , the We will prove (c) (a) by induction on / R Y only complete fan is the fan The correspond 6 R ( in pictured in Example L 3.1.11. ing toric variety is . This is homeomorphic to , the two-dimensional sphere, and hence is compact.
Now assume the statement is true for all complete fans + of dimension strictly Y . 6 be a less than / , and consider a complete fan in ? @ . Let sequence. We will show that has a convergent subsequence.
6
4 8
H , we may assume the seSince is the union of finitely many orbits 4 8 4 8 H H R quence . If P , then the closure of H 6 lies entirely within an orbit R in is the toric variety of dimension / Proposition 3.2.7. Since B D B 4 H 8 by Y is complete, 4 8 it is easy to check that the fan is also complete in ? H @ (Exercise 3.4.1). Then the induction hypothesis implies that there is a convergent
§3.4. Complete and Proper
139
4 8
subsequence in H . Hence, without loss of generality again, we may assume that 5 ! 6 . our sequence lies entirely in the torus Recall from the discussion following Lemma 3.2.5 that
5 . 5
,
I. *V4 ; J M 8
Moreover, when we regard as a group homomorphism Y , restriction yields a semigroup homomorphism 6 for any 6 . 4 3 hence a point in .
; M , then 7Q9 ; and
5 ? @ ;
A key ingredient of the proof will be the logarithm map 5 ; M of , consider the map as follows. Given a point by the formula
, R
I 4 , 8
4 8
. , This is a homomorphism and hence gives an element For more properties of this mapping, see Exercise 3.4.2 below. the most important property of 4 8 R 6 for some 6 . Y 5 us, . For satisfies . of
defined defined
I. *4 ; * 8 ? @
.
is the following. Suppose that a point . If , . 687:9 ; , then the definition
implies that
I 4 , 8 R K , 4 8 L 4 8 R 6 . Hence 4 , 8 . Thus we have which is since , . 6 7 and . proved that 4 , 8 4 8 . R 6 R (3.4.2) for all ,. 6 7 9-; 4 8 . ? @ . Since Y Now apply to our sequence, which gives a sequence R Y (3.4.1)
is complete, the same is true for the fan consisting of the cones Hence, by passing to a subsequence, we may assume that there is 6
4 8 . R 6 4 , 8 . By (3.4.2), we conclude that
6 for 6 . . . Y such that
for all for all ,. 6 7 9 ; . It follows that the are a sequence of mappings to the closed unit disk in . Since the closed unit disk is compact, there is a subsequence which converges to a point 4 3 . . You will check the details of this final assertion in Exercise 3.4.3.
Proper Mappings. The property of compactness also has a relative version that is used most often in the theory of complex manifolds.
Definition 3.4.2. A continuous mapping O "O ( 4 8 is compact in for every compact subset
is proper if the inverse image ! .
It is immediate that is compact if and only if the constant mapping from to the space R pt consisting of a single point is proper. This relative version of compactness may also be reformulated, for reasonably nice topological spaces, in the following way.
Chapter 3. Normal Toric Varieties
140
Proposition 3.4.3. Let O be a continuous mapping of locally compact first countable Hausdorff spaces. Then the following are equivalent: (a) (b) (c)
O
is proper.
4 4S8
is closed in for all closed subsets . , are compact. ' . 4 ' 8 . converges has a subsequence Exery sequence ' that converges in . such that O 4
O
is a closed mapping (that ( is, O ! ), and all its fibers O " 4 8 ,
Proof. A proof of (a) (a) (c).
(b) can be found in [33, Ch. 9,§4]. See Exercise 3.4.4 for
Before we can give a definition of properness that works for morphisms, we L need another criterion for properness. Recall from §3.0 that morphisms O L U and give the fibered product . Fibered products can also be defined for continuous maps between topological spaces. In Exercise 3.4.4, you will prove that properness can formulated using fibered products.
Proposition 3.4.4. Let O be a continuous map between locally compact Hausdorff spaces. Then O is proper if and only if O is universally closed, meaning
that for all spaces and all continuous mappings , the projection defined by the commutative diagram
&
/
N
/
is a closed mapping. In algebraic geometry, it is customary to use the following definition of properness for morphisms of algebraic varieties.
(
Definition 3.4.5. A morphism of varieties is proper if it is univer
sally closed in the sense that for all varieties and morphisms , the projection defined by the commutative diagram
&
/
/
is a closed mapping in the Zariski topology. A variety pt is proper. the constant morphism
is said to be complete if
§3.4. Complete and Proper
141
Example 3.4.6. The + Projective Extension Theorem [17, Thm. 6 of Ch. 8, §5] shows that for R , the mapping + P P
P ; . It follows that if ! is any affine + variety, the projection ; ; is a closed mapping in the Zariski By the gluing+ construction, it follows + topology.
pt is proper, so is a complete variety that the constant morphism is closed in the Zariski topology for all ,
(the prototypical complete variety). Moreover, any projective variety is complete (Exercise 3.4.5).
that this is not On the other hand, consider the morphism
pt . We claim J R proper, so is not complete. To see this, consider and the diagram pt
.
The closed subset ( under . Hence
/
/ pt
354 ' R 8 ! does not map to a Zariski-closed subset of ( is not a closed mapping, and is not complete.
Completeness is the algebraic version of compactness, and it can be shown that a variety is complete if and only if it is compact in the classical topology. This is proved in Serre’s famous paper G´eom´etrie alg´ebrique et g´eom´etrie analytique, called GAGA for short. See [92, Prop. 6, p. 12]. The Properness Criterion. Theorem 3.4.1 can be understood as a special case of the following statement for toric morphisms.
6
6
Theorem 3.4.7. Let be the toric morphism corresponding to a ? ? Z that is compatible with fans Y in ?A@ and Y Z in ? @ Z . homomorphism Then the following are equivalent:
is proper in the classical topology (Definition 3.4.2). is a proper morphism (Definition 3.4.5). ! # 4 8 exists in , then # 4 8 exists in and
(b)
6
(a)
6
(d)
6
(" . 4 Y? Z 8 Y R @
(c) If
6
)
)
6
6
.
.
Proof. The proof of (a)
(b) uses two fundamental results in algebraic geometry.
First, given any morphism of varieties O and a Zariski closed subset ! , a theorem of Chevalley tells us that the image O 4 4