1 Preliminaries
In this
chapter of preliminaries
we
review from
[3], [6], [17]
and
[31]
some con-
cepts and wel...
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1 Preliminaries
In this
chapter of preliminaries
we
review from
[3], [6], [17]
and
[31]
some con-
cepts and well-known results about birational maps of surfaces and weighted clusters and derive maps in
chapters
some
applied to plane Cremona proofs of these consequences have been
consequences that will be
2 and 4.
Only
the
included. this
Throughout
irreducible surface
chapter
over
[3]
notations of
with
surfaces,
Ll
Blowing-ups
we
by surface a smooth projective complex numbers C. When dealing adopted.
shall
mean
the field of the are
Definition 1. 1. 1 Let S and S' be two surfaces. A rational map !P : S --+ S' is a morphism from an open subset'U of S to S' which can not be extended to any
points to
open subset. The of 4i. A birational, map P
larger
some
points :
S'--+ S' is
non-empty open subset V of S is
Remark 1.1.2 Since
we
S
in F a
an
-
U
are
called
fundamental
rational map whose restriction
isomorphism.
will deal with rational maps that are birational maps are dominant and their composition
,between irreducible surfaces, these
always makes Let D be
sense.
a
divisor
on a
surface S. The set of all effective divisors
on
S
linearly equivalent to D will be denoted by I D 1. 1 D I can be identified with the projective space associated to the vector space HO (Os (D)),. where Os (D) is the invertible sheaf corresponding to D. As long as no confusion may result, we use
the
same
notation for
a
divisor and its class in Pic S.
subspace C of IDI is called linear system on S. complete if C ID1. The dimension of C is by definition its dimension as a projective space. We say that a generic element of C has a property P if elements in a non-empty Zariski-open set of the projective space C have the property P. We say that C has a curve G as a fixed component if every divisor in C contains G. The fixed part of C is the biggest divisor F that is contained in
Definition 1.1.3 A linear C is said to be
every element of C. Then the linear
system C
M. Alberich-Carramiñana: LNM 1769, pp. 1 - 28, 2002 © Springer-Verlag Berlin Heidelberg 2002
-
F has
no
fixed part.
1 Preliminaries
Let P
:
S
--+
F of !P form
S' be
a
birational map of surfaces. The fundamental points points of S Q3] IIA).
finite set of
a
Definition 1.1.4 Let C be
an irreducible curve on S. Denote by 4i(C) the -P(C F) in S', which will be called the image of C. The direct 0 if -P(C) is a point, or P,, (C) image P,, (C) of C is either !P,, (C) -P(C) if -P(C) is a curve. We define P(D) and !P.(D) for any divisor D on S by
closure of
-
=
=
linearity. Let D be
4i*(D)
a
divisor
on
S'. The pull-back of D by P is denoted by
and is called the total
We shall maps, the
see
transform of
(cf. [31 IIA).
D
that birational maps are composites of elementary birational to which the rest of this section is devoted.
blowing-ups,
Definition 1.1.5 Let S be
a surface, and let p be a point in S. We denote by H : 3 -4 S. The restriction of the morphism H to H-'(S jp}) is an isomorphism onto S fpj, and E := H-1(p) is isomorphic to P1, is called the exceptional divisor of H and can be identified
the
blowing-up of p
on
S
-
-
with the tangent directions For
[6]
3.1
a or
on
S at p.
construction of 1Y and its basic
[31]
Definition 1.1.6 Let H and consider
denoted
by 1
properties
see
for instance
[3] IIJ,
V.3.
S be the
blowing-up
of
a
point
p
on
S,
S. The image of C by the birational map H-1 is and called the strict transform of C (after blowing up p).
a curve
C
on
Definition 1.1.7 Let p, be
a
point
in
a
surface
S,
let
Os,p
be the local
ring
of S at p, and let Mp be the unique maximal ideal of Os,p. Suppose, C is a curve on S, and f E Os,p is a local equation of C at p, then the multiplicity of C at p is the integer ep(C-) for which the relation f E holds. Clearly ep(C) > 0 if and only if p belongs to C. Lemma 1.1.8
S
of form
an
([3] 11.2, [6] 3.2.1)
irreducible
curve
C
on
=
The total
S that has
i
+
A4ep(C) -,A4ep(C)+l
transform after blowing up p on multiplicity ep(C) at p has the
ep(C)E
.
S be the blowingProposition 1.1.9 ([3] 11.3, [31] V.3.2, V.3) Let H up of a point p on a surface S, and let E be the exceptional divisor of 17. 1.
There is
an
isomorphism Pic S E) Z
(D,, n)
Pic'3 1Y
+ nE
.
1.1
2. Let C and D be divisors
S. Then
on
ZT 1
=
C
ZT-E
=
0,
=
_1
-
E 3.
Blowing-ups
Projection formula: let C be
a
2
divisor
-
D
.
S and let D be
on
a
divisor
on
3.
Then
C
ff rs div(w) 2-form on S, then =
is
a
-
D
=
C
-
canonical divisor
(H.D) on
the canonical divisor
r.y
=
Definition 1.1.10 Let 0 be
div a
(11* (w))
point
=
S, 'where w is a meromorphic 3 that corresponds to rS by
on
H* rs + E
in S. The
exceptional divisor
E of blow-
S will be called the first infinitesimal neighbourhood of 0 on S and its points will be called the points in the first infinitesimal neighbourhood
ing
up 0
on
of 0 (on S). If i > 0, we may define by induction the points in the i-th infinitesimal neighbourhood of 0 (on S) as the points in the first infinitesimal neighbourhood of some po'int in the (i 1)-th infinitesimal neighbburhood of 0. In the sequel we will often drop the adjective infinitesimal by saying just neighbourhood instead of infinitesimal neighbourh'ood. The points which are in the i-th neighbourhood of 0, for some i > 0,- are also called points i7ifinitely near to O.'Sometimes the points in S will be called proper points in order to distinguish them from the infinitely near ones, as the word point will be used for both kind of point. Let p, q be two points in S proper or infinitely near. We will say that p precedes q and write p, < q if and only if q is infinitely near to p. We will write p < q if q is equal or infinitely near to p. The relation < is a partial ordering and will be called the natural ordering of the infinitely near points. -
only concerned with the blowing-up of a single point. blowing-up of a subset of proper or infinitely near points in the surface S, which essentially consists of the successive blowingups of all the points, provided that after each blowing-up Sj --+ Sj_1 we identify the points not yet blown up to their corresponding ones on the surface Until
Now
we
now we were
will deal with the
Si Definition 1.1.11 A cluster in
a
surface S is
a
finite set K of proper
or
infinitely near points in S, so that, for each point p E K, K contains all the points preceding (by the natural ordering) p. The proper points of K are called the origins of the cluster. (Notice that these clusters are union of finitely many clusters in the sense of [6] 3.9). A 'subeluster K' of a cluster K in S is a subset of K which is also a cluster in S. By a maximal point in K we shall mean a maximal point in K relative to the natural ordering on K if no other ordering is mentioned.
1 Preliminaxies
4
Definition 1. 1. 12 A is
an
called
a
pair IC
=
(K, #),
map, will be called
arbitrary
a
where K is
weighted
system of virtual multiplicities for (the
K will be called the
a
cluster and A
:
K
--+
Z
cluster. The map [t will be points,of) the cluster K and
underlying cluster of IC. We will usually multiplicity of the point p.
write I-Lp
=
It(p)
and call pp the virtual Let K be
cluster with
0, in a surface S. We. denote origins 01, points in) K by 1TK : SK -4 S. For a detailed construction of HK and its basic properties see for instance [6] 3.5 and 4.3.
the
a
blowing-up of (all
We outline below
.
.
.
,
the
only the
main features.
Definition 1.1.13 An
ordering -- on the points in a cluster K is admissible only if for any pair p, q E K so that p :5 q, we have p : q. That is, an admissible ordering is a refinement of the natural ordering. if and
Fixed
an
admissible total
ordering -
on
K, then HK
is the
composite
of the sequence of blowing-ups of the points in K following this admissible ordering, and SK is the surface obtained from S after these blowing-ups. This construction is
essentially unique (that is, if SK' is the blowing-up of the points in K following another admissible ordering, then there is a unique Sisomorphism from SK onto SK), and hence all the notions related to HK that will be introduced from now on are independent of the admissible ordering used for
defining
them.
The restriction to
Hil(S
-
101,...,0,1)
of the
morphism HK
is
an
.
isomorphism onto S 101,..., 0,}. The origins of points of the birational map ff, ' : S --+ SK -
Definition 1.1.14 Consider
C
K
are
the fundamental
S. The pull-back of C
by IYK is transform of C (after blowing up K). The direct image of C by ITil is denoted by I K (C) and is called the strict transform of C (after blowing up K). denoted
Both
by e
=
ff, (C)
transforms,
OK
and strict transforms of
a curve
on
and is called the total
and
I K,
may be also obtained as the iterated total the blowing-ups composing LIK.
C, respectively, by
Consider the sets
Kp=jqEK:q-- 0 As the components of a vector =
-
in
a
basis
claim.
are
unique,
we
infer that ap
El
Suppose S
projective plane, and let
is the
PyK, Ep",
_K
-K
are
to these bases is
.
.
,
of
a
PK-1
a
line in S.
1.1.26
-K'...je EP
P,
matrix of the
identity
map
PK-1
=
be the inverse
(aPq)(p,q)EKxK
Suppose
is
an
admissible
has all its entries above the
ordering
Idpi, sK relative
are
of on
the
proximity
K.
diagonal equal to zero, those diagonal are all
to one.and those below the
all
on equal diagonal non-negative. 2. Let p, q E K, p -< q If q is a free point, and hence single point r E K, then
the
By
-
cluster K.
The matrix
H be
and
Ep-',,.
SK and the PK*
Lemma 1.1.32 Let
1.
.
two bases of Pic
matrix
for each p E K, and hence the
bp,
=
-
-
a
q
=
ap"
.
it is
proximate
to
a
1.1
3. Let p, q E
just
two
K, p -< q. If q points r, and r2
is
K, then
apq 4. Let
PROOF: Let
PKPK_1
=
p -< q. Then q is
K,
p, q E
aq
denote the
a" + 12 ap P
infinitely
q-th
row
I and the definition Of PK
=
a
it
q
11
satellite point, and hence it is proximate to
a
in
Blowing-ups
+
q
near
to p
if
and
PK-1.
of the matrix
easily give
only if apq The
> 0.
equality
the relation
1: j ,qar rEK 'r - 0. We
induction
aq > 1 > 0 P
according
to assertions
1,
-
2 and 3.
Assume that q lies on the n-th neighbourhood of p. Let r E K be the point whose first neighbourhood q lies on. Then r lies on the (n I)-th neighbourhood of the point p, and by induction hypothesis > 0. Now, -
ap'P
owing
to assertions
1,
2 and
3,
we
obtain
aq> a' > 0, P P -
as
wanted.
Observe that,
according to assertion 1 if apq is not positive, then apq van going to see now that if q is not infinitely near to p, then apq 0. Assume first that q is a proper point. Then from (1.1) we have ishes. We
are
=
a
q
q
and hence
a
Assume
=
0.
that q lies on the n-th neighbourhood of a proper point one of the points which q is proximate to, then r lies the m-th neighbourhood of the point 0, with m < n 1. By induction
0 E K. If on
q P
now
r
E K is
-
hypothesis apr
=
0.
Hence, owing
to assertions 3 and q
and
we are
done.
n
=
0
4,
we
obtain
1 Preliminaxies
12
Example
1.1-33 Let
PK be the proximity matrix appearing in 1.1.29. Then 100000000
110000000 211000000 311100000
P-1 K
110010000 110001000
220010100 321000010
321000011) Definition 1.1.34 The intersection matrix NK of the cluster K is square matrix defined p-th column.
K by taking EP ..j qK t
An easy computation Lemma 1.1.35
NK
=
as
the entry
the
on
q-th
a
K
row
x
K
and
using 1.1.26, 3 gives the relation:
-PtKPK-
Definition 1.1.36 Take C
a curve on
S. Let
us
write
e=ff+E vp (C)-K EP' PEK
Each
vp(C)
is
a
non-negative integer which will be called the effective p-value (O)PEK and we call it the vector of effective values
of C. We put VK (C) = (VP of C at the points of K.
Clearly
from the
the
definition,
proximity
matrix of K relates
multiplici-
ties and values:
Lemma 1.1.37 For any
curve
VK
Definition 1.1.38 If C is in S with
C
(C)
S,
on
=
PK'eK(C)
a curve on
S and Q
cluster K, we say that the the surface SK
underlying
if the divisor
on
=
(K, v)
curve
is
goes
a
weighted
cluster
(virtually) through
K
CQ
VPEP PEK
is effective.
weighted
OQ
is called the virtual
cluster Q.
transform of
the
curve
C relative to the
1.1
If
points
eK(C)
is the vector of effective
in the cluster
K, then. the
Blowing-ups
13
multiplicities of the curve C at OQ can be written in
virtual transform
the the
form
OQ
=
I K
+
_K E UP(C) EP'
PEK
where the vector
UKM
=
(UP(QpEK
UK (C)
Definit on
1.1.39 If
we
PK1 (eK (C)
=
have the
say that the
we
multiplicities equal
curve
C goes
=
-
as
V)
of vectors
equality
eK(C) then
is obtained
V)
through the weighted cluster Q with effective
to the virtual
ones.
Example 1.1.40 Keep the notations of example 1.1.22. Figure 1.2 shows Enriques diagram of the cluster K, in which the (effective) multiplicities of C at the points of K have been indicated. We have eK (C) (2, 1, 1), the
=
A
2
140 P2
1.2.
Fig.
VK(C)
(2,3,6)
Enriques diagram of the CUSPY2
.
X3
at the
origin.
and
PK
1
0
0
-1
1
0
-1 -11
(K, v),
Put Q 4 x
at the
eK (D)
=
with
1.
The
curve
2.
The
following The
=
eK
and consider the
as
=
Lemma 1.1.41
a)
(C),
planar tacnode D y2 _K _bK + Epl, origin. Then D goes virtually through Q andbQ (2, 2, 0) and VK (C) (2, 4, 6). v
With the notations
b) UK(C)
above:
C goes virtually through Q if and only three assertions are equivalent:
curve
virtual
as
C goes
ones. =
0.
through Q
with
if
UK
(C)
> 0.
effective multiplicities equal
to the
1 Preliminaxies
14
c)
The virtual
C
point.
and the strict
a
curve
going
ep(C)
Then
>
(K, v)
through
i
K of
the
curve
and let p E K be
a
proper
vp.
PROOF: Assertions 1 and 2 follow assertion 3.
transform
equal.
are
3. Let C be
transform OQ
directly
from the definitions. Let
us see
1.1-41
By
up(C)
0
curve,
values,
0,
multiplicities equal
to the virtual
ones
if
VK(C) The
1C excess
P
of
a
Lemma 1.1.44
IC
P
=
Next result focuses is
be
weighted
cluster IC
(K, M)
at
a
point
p E K
(1.1.24)
be written in the form
can
a a
cluster in line in
Ip2'
p2.
AtPKIp on
where
the
an
=
WPtK PKIp
case
=
2 projective plane P and ordering has been fixed. Let
that S is the
admissible total
-WNKlpK H
1.1
Lemma 1.1.45 Let C be
SK linearly equivalent
a curve on
=--K
Blowing-ups
15'
to the divisor
-K
aoH
1: apEp
-
PEK
with ap E then D goes
=
Z, and let Ka be the weighted cluster IIK,, (C) is
a
curve
on
and its virtual
virtually through ICa
PROOF: The effective divisor C
D
=
F-pEKypEp'K,
eK(D)
ao >
i4 0,
0),
D
to C.
,
degree
of the
curve
D
on
following
p2 is
column
K
on
aK
=
7K
=
bK where
ao
as
1.1.26. Consider the
by
ao,
If
j5K
with -yp > 0. The
H, which equals C,
-
vectors'indexed
ao (in particular transform br-' is equal
be written
can
C=E + where E
(K, laPIpEK).
p2 of degree
=
(aP)PEK
('YP)pEK PK1 (eK (D)
is the vector of effective
points of the cluster
-
aK)
multiplicities of the
curve
D at the
K. We have
E
=
C
bK
_
E
-==K
(ep (D)
-
ap) EP
pEK
which
implies bK
Hence, according
to
1.1.41,
,DlCa
f)K
=
D goes
^IK > 0
virtually through the weighted
cluster IC a
and +
_K 1: bAt
_K
=D
+E=C.
pGK
The
infinitely
tiplicity
of two
near
Theorem 1.1.46 curves
on
a
points give a geometrical idea of the intersection mula point by means of a formula due to A Noether:
at
curves
(Noether's formula, [6] 4.1.3)
surface S
multiplicity [C D]o
is
points infinitely
to
-
near
and let 0 be
a
proper
point
finite if and only if C 0, and in such a case
[C D]o -
=
1: ep(C)ep(D)
running for p infinitely
near
and D
to 0.
be two
The intersection
and D share
P
the summation
Let C in S.
finitely
many
1 Preliminaries
16
Even if we do not know the effective
but
only
its virtual
intersection
multiplicities multiplicity:
at
a
multiplicities of one of the two curves, cluster, we have a useful bound for the
Proposition 1.1.47 (Virtual Noether's formula, [6] 4.1.3) Consider a (K, v) in a surface S. Assume C is a curve on S and weighted cluster IC 0, are the origins of K. If D is a curve on S going through K, then 01, =
S
E [C D]O, -
E ep(C)vp
!
i=1
.
p(=-K
Example 1.1.48 Consider the cusp C and the tacnode D appearing in example 1.1.40. Using Noether's formula we compute the intersection multiplicity of C and D at the origin p
[C D]p
=
-
while
ep(C)ep(D)
(C)ep, (D)
applying virtual Noether's formula
[C D]p -
and
+ ep,
we
2
>
Ili
see that in this case
+
2
1/ l
+
we
2
I'P2
=
=
4 + 2
6
obtain 4+ 1 + 1
=
6
the bound is reached.
Definition 1.1.49 Let C be
a
linear system
on
S without fixed part. The
of base points of C is a consistent weighted cluster K(C) defined in the following way. Start by taking the proper points 01,. Os E S so that every divisor of C contains them. For Oi, I < i < s, take the virtual multiplicity v(Oi) equal to the minimal multiplicity at Oi of the divisors in C. Fix i E f 1, s}. Then discard from C the divisors with multiplicity at than Oi bigger v(Oi), and call C, -the family of the remaining ones. If these divisors do not share any point in the first neighborhood of Oi, then our subcluster with origin Oi is just Oi with virtual multiplicity V(Oi). Otherwise take all the points that the divisors in C, share in the first neighborhood of Oi, each point p with virtual multiplicity equal to the minimum of the multiplicities at p of the divisors in C1. Again discard the divisors whose multiplicities are not the minimal ones and look for the points the remaining divisors share in the first neighborhoods of the former ones, and so on. This process is repeated for each 1 < i < s. The procedure clearly ends after finitely many steps, as
weighted
cluster
-
.
.
.
,
V(P)2 PEK(C) for C E C.
< C. C
-,
1.2
(Bertini's theorem)
Theorem 1. 1. 50
fixed
out
Let C be
Weighted
clusters
linear system
a
on
17
S with-
part. Then
1. A
generic element of C goes through IC(C) with effective multiplicities equal to the virtual ones and has no singular points outside of IC(C).
IQC)
2.
is consistent.
3. Either C is
composed of
curves
in
a
pencil,
or a
generic element of C
is
irreducible. PROOF: Notice that in the definition of
divisors in
IC(C)
at each
step
we are
discard-
Zariski-closed set,of the
projective space C, hence a generic element of C goes through IC(C) with effective multiplicities equal to the virtual ones and so, according to 1.1.42, IC(C) is consistent. The rest of assertion
ing
[6]
1 is
a
7.2.
Froin'the definition of
system CK theorem
---:
I OK
:
linear systems
on
weighted
C E C I has
([31]
pencil, or a generic element implies assertion 3. M in
a
1.2
Given
a
weighted
cluster IC
admissible total
an
points.on SK
Of
is irreducible and
CK
non-singular. This
=
(K, v)
ordering,
in
S, with K
=
Jpi,
defines
a
(- vpI EP-.
(-UK)* OSK
=
p, I written
the ideal sheaf -K
WIC
ideals
4eIinear
clusters
Weighted
following
cluster of base
fixed part and no base points. By Bertini's JII.10.9.1) either CK is composed of curves
no
zero-dimensional subscherne of
-
-
-
-
-
vp,
P,
)
S, and the stalks of RIC
are
complete
app.4) in the stalks of Os. Conversely, if I is a coherent sheaf of ideals on S defining a zero-dimensional scheme and whose stalks are complete ideals, then there is a weighted cluster IC in S so that I 'RIC (see [6] 8.3.7). A curve on S contains the scheme defined by IC if and only if it goes virtually through IC (1.1-38). If p E S is an origin of the cluster K, then
([51],
v.II
=
the stalk of RIC at p, is
W)c,p
=
If
E
0S,p: vp(f) : Vp}
where vp (f) is the vector of effective values of the germ of curve f = 0 at the subeluster K(p) Iq E K : p < q} C K and Vp is the system of values of the weighted subcluster of IC whose underlying cluster is K(p), otherwise =
RIC&
=
os'p.
Definition 1.2.1 Two
only
if RIC
=
W)C,.
weighted
clusters IC and IV in S
are
equivalent if and
1 Preliminaxies
18
It follows
directly from
Lemma 1.2.2 Let IC
the definitions: and 10
(K, v)
=
S whose values satisfy V
Then
> T.
(K, v')
=
we
have
for
be two
weighted clusters
in
S the inclusion
of
any p E
stalks
RIC,& Definition 1.2.3 Let /C the order
(K, v)
=
UK&
C
be
weighted
a
cluster
on
S. We define:
of singularity of IC
VP(VP
E
-
2
PEK
the virtual codimension of K
c
VP (VP +
(K)
2 pEK
self-intersection of IC
and the
IC. K
2
E
=
pEK
Clearly, IC Given
a
/C'=
-
J(]C)
+
c(IC)
(1-3)
IC in p2 and
weighted cluster
system of all the plane curves of degree HI (]?2, -HK 0 Op2(n)). by fK (n)
n
a positive integer n, the linear going through K will be denoted
=
The number of
presents
to the
independent conditions that aweighted of degree n is defined as
cluster /C in p2
curves
In (n + 3)
-
2
Definition 1.2.4 If
(1.4)
is
an
imposes independent conditions
dim tK (n) :5 c(IC)
(1.4)
.
equality, we say that the weighted cluster IC to the plane curves of degree n. The integer
1 2
n(n
+
3)
7
C(IC)
will be called virtual dimension of t1c (n) and will be'denoted
By (1.4)
we
-vdim and the curves
equality degree
of
by vdim. f1c (n).
have
holds if and n.
tjc(n)
only
V2 ,
1.2.2
by
have for any
we
'HIC1,X
C
E
x
PI the inclusion of stalks
' IC2,X
and hence the desired inclusion of linear systems. 11 Next lemma shows how the excess, the virtual codimension and the orsingularity behave by increasing values, following directly from the
der of
definitions.
-
Lemma 1.2.8 Assume that the
weighted cluster IC'
from IC (K, v) of p, by n of excesses at p is by increasing pp' pp (rp + 1)n and at q 54 p, is either pq Pq -n if one of the points p or q is maximal among the points in K that are proximate to the other, or 0 otherwise. The variation of virtual codimension is is pq pq =
-
-
comes
=
units. Then the variation
the value
=
-
=
n
c(IC') and the variation
-
c(IC)
=
2
(2pp
+ 2 +
of order of singularity
(n
-
1)(rp
+
1))
1)(rp
+
1))
is
n
6 (IC,) _6 (IC)
.
2
(2pp
-
2 +
(n
+
of the cluster IC, and rp
where p,, is the excess at p, in K proximate to p.
is the number
of points
Enriques Q26] IV.II.17) called unloading (see weighted cluster IC (K, V) in S (K, v') is consistent and gives a new system of multiplicities v' so that IC' equivalent to IC. At each step of the procedure some amount of multiplicity /C < 0 from the points is unloaded on a point p, E K, at which the excess P that are proximate to it. Let us present it in terms of increasing values. There is
[5]
or
a
procedure
[6] 4.6)
that from
due to a
non-consistent
=
=
I
Definition 1.2.9 Let IC X
P
< 0 that
is, according
=
to
(K, v)
be
n as
the least
integer
weighted
cluster and
assume
that
1.1.44,
VtNKI-p Define
a
so
-n(rp
> 0
-
that +
1)
+
VtNK 1p
< 0
with rp the number of points in K proximate to p. the value of p by n.
Unloading
on
p, is
increasing
Assume IC (K, v) is a non-consistent weighted and, inductively, as far as IC'-' is not consistent define K' from )C'-' by unloading on a suitable point. Then we have: Theorem 1.2.10
cluster. Put ICO
=
([6] 4.6.2)
IC
=
1.2
There is
1.
K
as
that /Cm is consistent, has the
an m so
IC and is
Weighted
equivalent
same
clusters
21
cluster
underlying
to it.
only consistent weighted cluster which is equivalent to )C and underlying cluster. In particular, it does not depend on the of the points on which the unloadings are performed.
2. Km is the
has the choice
I et
same
introduce
us
Definition 1.2.11 tame
unloading.
K
as
rp!+_1
with 1 >
Example a
case
-1
=
of
unloading that
can
be found in
[6]
4.7.
point of excess equal to -11 will be called 1, unloading the value is increased in n and hence n is the least integer so that n > rp+l
Unloading
Note that in
-VtNKIp
=
P
special
a
a
on a
tame
(1-2.9). Figure 1.3 shows a sequence of three unloading steps from weighted cluster-to its equivalent and consistent one. Obsteps 1 to 2 and 2 to 3 are tame unloading, while the step 3
1.2.12
non-consistent
serve
that the
to 4 is not tame.
0 0
0
1
A-**
0 0
R3
R2
Fil
0
0
%0
0
0
F4]
Fig. 1.3. A sequence of unloading steps from a non-consistent system of multiplicities (left) to the,,, corresponding consistent one (right). The black indicate the points on which multiplicities are unloaded.
Proposition 1.2.13 ([6], 4.7.2) Assume from IC (K, v) by unloading on p E K. =
C(IC') and the
equality holds if and only if
Lemma 1.2.14 Assume that the
by
tame
unloading
in K. Then
on
:5 the
that the
weighted cluster IC'
virtual arrows
comes
Then
c(IC)
,
unloading
weighted
is tame.
cluster IC'
p, E K. Let rp be the number
comes from IC (K, v) of points proximate to p =
22
1 Preliminaries
J(r) and
if p,
is
a
-
non-maximal point
rp
(1. 1. 11),
J(r) ,
PROOF:
(1.5)
According to 1.2.11, pprby substituting in 1.2-8.
=
follows
then
Jpq
>
(1.5)
-
.
-1 and
n
=
1, after which equality
If p is non-maximal then rp > 1 and
hence the claim. 1:1
Remark 1.2.15 Let IC
(K, v) be a weighted cluster having non-negative multiplicities that gives rise to the consistent weighted cluster Q by tame unloading. The tame unloading steps may be performed in such a way that the intermediate weighted clusters have non-negative virtual multiplicities. Indeed, at each step, first drop successively maximal points with virtual multiplicity zero. Once there is no one of these, unload on a point that is maximal among those of virtual multiplicity zero if any. Since the amount unloaded at each step equals one (1.2.11), this guarantees that no multiplicity becomes negative. =
virtual
Lemma 1.2.16 Let IC
clusters in S
so
=
(K, v)
and V
c(/C') d(r) IV IV -
Furthermore, 1. IC
2.
=
-
be two consistent
weighted
the
following three
>
c(IC)
(1-6)
>
6(IC)
(1.7)
> Ic
Ic
-
assertions
(1-8)
are
equivalent:
V.
c,()C)
3. )c
(K, v')
=
that T' > -F. Then
IC
c(IC'). IV
-
V.
PROOF: Let p E S be
origin of
an
subclusters of IC and V whose
K(p) By 1.2..2, Wlcp,p
f1Cj}j=0'... np
D
=
K. Let
u pderlying
fq
and
)p
1CP'
be the
weighted
cluster is
E K: p,
0.
=
EI
Proposition 1.3.12 Keep the notations and hypotheses of a hyperplane section in S' C pn, and take p, E K'. Then ==K'
EP_
1.3.10. Let H be
-_L'
H
-
> 0.
-K'
Furthermore, PROOF:
0
Ep-
if and only if p,
=--L'
K,
then E
L' '
H
=
h*
P
If p, E K- K
=
(--K E_ P)_ h*
1.3.13
-:-L'
EP_
HI
3.E q
following
surfaces
-K'
L'
Keep the
K =
ITK'
Y and
> 0 > 0
for
for
PROOF: It follows
SK
H
=
ILV > 0
substituting
in
(1.11) gives
--L' -
H
conditions
all the
L ,
P
0 and
o.
notations and
are
E
K
(
are
isomorphic.
points
p E K.
all the points q c L'.
directly from
Ei
hypotheses of 1.3.10. a hyperplane section equivalent:
section in S C F' and let H' be
Then the three
2.
H > 0
(_HL)
f,, FP
L'- L, then
Ep-
The
K.
K
(EP-
H
-K'
1.
-
1.1.26 and 1.3.11
by
K'
Corollary hyperplane
KI
By the projection formula (1.1.26, 6)
EpIf p E
E
1.3.12. El
Let H be
a
in S' C Pn.