A bound on the plurigenera of projective varieties

manuscripta math. 112, 391–401 (2003)

© Springer-Verlag 2003

Vincenzo Di Gennaro

A bound on the plurigenera of projective varieties Received: 7 March 2003 / Revised version: 26 August 2003 Published online: 20 October 2003 Abstract. We exhibit a sharp Castelnuovo bound for the i-th plurigenus of a smooth variety of given dimension n and degree d in the projective space Pr , and classify the varieties attaining the bound, when n ≥ 2, r ≥ 2n + 1, d >> r and i >> r. When n = 2 and r = 5, or n = 3 and r = 7, we give a complete classification, i.e. for any i ≥ 1. In certain cases, the varieties with maximal plurigenus are not Castelnuovo varieties, i.e. varieties with maximal geometric genus. For example, a Castelnuovo variety complete intersection on a variety of dimension n + 1 and minimal degree in Pr , with r > (n2 + 3n)/(n − 1), has not maximal i-th plurigenus, for i >> r. As a consequence of the bound on the plurigenera, we obtain an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef.

Continuing a previous work on the plurigenera of projective surfaces [D1], in the present paper we prove a Castelnuovo bound for the plurigenera of projective smooth varieties of any dimension. Fix integers n, r and d, and let V(n, r, d) be the set of all projective, irreducible, smooth and nondegenerate varieties V of dimension n and degree d in the projective space Pr . For any integer i ≥ 1 put
 P (n, r, d, i) = sup pi (V ) : V ∈ V(n, r, d) , (0.1) where pi (V ) denotes the i-th plurigenus of V . In the case i = 1 Harris [H] proved that     m m P (n, r, d, 1) = (r − n) +
, n+1 n where m and
are defined by dividing d − 1 = m(r − n) +
,

0 ≤
< r − n,

(0.2)

and, when d > n(r − n) + 1, he classified the varieties with maximal geometric genus P (n, r, d, 1), the so-called Castelnuovo varieties. In this paper we compute P (n, r, d, i) and classify the varieties V ∈ V(n, r, d) which achieve the bound, at least when r ≥ 2n + 1, d >> r and i >> r. More precisely, we will show the following V. Di Gennaro: Universit`a di Roma Tor Vergata, Dipartimento di Matematica, Via della Ricerca Scientifica, 00133 Roma, Italia. e-mail: [email protected] Mathematics Subject Classification (2000): Primary 14J99; Secondary 14N99

DOI: 10.1007/s00229-003-0415-z

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Theorem. Let V ⊂ Pr be a projective, irreducible, smooth and nondegenerate variety of dimension n ≥ 2 and degree d (i.e. V ∈ V(n, r, d)). Denote by pi (V ) the i-th plurigenus of V . Assume r ≥ 2n + 1, d > 165n5 (r − n)3 and i > 2n3 (r − n). With the same notation as in (0.2), define
0 (n, r) = [(n + 3)(r − n) + 2(n − 1)] /2(n + 1), and

(0.3)



p(n, r, d, i, a) =

   i(m − n + a) + n (i − 1)(m − n + a) − 1 + (0.4) n n     i(m − n + a) + n (i − 1)(m − n + a) − 1 + i −(i −1) n n   (m − n + a)(r − n) × +
− 1 − a(r − n) . n+1

We have: (i) if 0 ≤
<
0 (n, r), then pi (V ) ≤ p(n, r, d, i, 0); (ii) if
0 (n, r) <
, then pi (V ) ≤ p(n, r, d, i, 1); (iii) if
=
0 (n, r), then pi (V ) ≤ max{p(n, r, d, i, 0), p(n, r, d, i, 1)}. All previous bounds are sharp. Therefore, in the appropriate ranges, they are equal to P (n, r, d, i) (see (0.1)). Moreover: • in case (i) we have pi (V ) = P (n, r, d, i) if and only if V is residual to r−n−1−
n-planes in the complete intersection of a rational normal scroll of dimension n + 1, whose singular locus has dimension < n − 1, with a hypersurface of degree m + 1; • in case (ii) we have pi (V ) = P (n, r, d, i) if and only if V is residual to 2(r − n) − 1 −
n-planes in the complete intersection of a rational normal scroll of dimension n + 1 whose singular locus has dimension < n − 1, with a hypersurface of degree m + 2; • in case (iii) we have pi (V ) = P (n, r, d, i) if and only if V is as in the maximal case of (i) (only when p(n, r, d, i, 1) ≤ p(n, r, d, i, 0)), or as in the maximal case of (ii) (only when p(n, r, d, i, 0) ≤ p(n, r, d, i, 1)). In case (i), the varieties with maximal i-th plurigenus are Castelnuovo’s, i.e. of maximal geometric genus. This is not true in case (ii) (and partly in case (iii)). For example, a Castelnuovo variety complete intersection on a variety of dimension n + 1 and minimal degree in Pr , with r > (n2 + 3n)/(n − 1), has not maximal i-th plurigenus, for i >> r. The new varieties V appearing in the classification are not arithmetically Cohen-Macaulay (on the contrary Castelnuovo varieties are). However they satisfy the property hj (V , OV ) = 0 for any 1 ≤ j ≤ n − 1. In particular their arithmetic genus is equal to the geometric one. Moreover, their canonical bundle is big and nef. Notice also that, from the Theorem, it follows that P (n, r, d, i) =

  d n+1 n+1 n+1 − (i − 1) i + O(d n ). (n + 1)!(r − n)n

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In case (iii) (i.e. when
=
0 (n, r)), one can prove that when d >> i then p(n, r, d, i, 1) < p(n, r, d, i, 0). The line of the proof of the Theorem is the following. Using Castelnuovo-Halphen’s Theory, we reduce the proof to an analysis of certain varieties V ∈ V(n, r, d) lying on varieties of dimension n + 1 and minimal degree r − n in Pr (see Corollary 4). Using Kawamata-Viehweg Vanishing Theorem, one can compute the plurigenera of such varieties, and sees that they can be expressed by means of the function p(n, r, d, i, a) defined in (0.4). Then one may conclude the proof of the Theorem by determining the maximal values of the function p(n, r, d, i, a) with respect to the variable a (see Lemma 6). The classification given in the Theorem is not complete because of the hypotheses r ≥ 2n + 1, d > 165n5 (r − n)3 and i > 2n3 (r − n). The first one is necessary for constructing smooth varieties of any dimension n and any degree d in Pr , at least for d >> r. But recall that any smooth variety of dimension n can be embedded in P2n+1 . In the case of surfaces (i.e. when n = 2), and in the case of threefolds (i.e. when n = 3), we are able to analyze also the case r ≤ 2n. We refer to [D1] for the case n = 2 and r = 4. In the case n = 3, using a similar analysis as in the proof of the Theorem, one can prove that if V ∈ V(3, r, d), 5 ≤ r ≤ 6 and
= 0, then pi (V ) ≤ p(3, r, d, i, −1). If 5 ≤ r ≤ 6 and
= r − 1 then pi (V ) ≤ p(3, r, d, i, 0). The bounds are sharp and hold for d > 20r 2 and any i ≥ 2. When
= 0 then pi (V ) = p(3, r, d, i, −1) if and only if V is homologous, on a rational normal scroll of dimension 4, to a 3-plane plus a complete intersection with a hypersurface of degree m. When
= r − 1 then pi (V ) = p(3, r, d, i, 0) if and only if V is a complete intersection of a rational normal scroll of dimension 4 whose singular locus has dimension < 2 (or, only in the case r = 5, of a quadric) with a hypersurface of degree m + 1. The case (r,
) = (6, 1) escapes to the classification, because in this case any threefold on a fourfold of minimal degree is singular. The assumption d > 165n5 (r − n)3 and i > 2n3 (r − n) is essential in our proof of the Theorem. We need it for using Castelnuovo-Halphen’s Theory (Proposition 3 and Corollary 4) and for several numerical estimates in the proof of Proposition 3, Lemma 6 and at the end of the proof of the Theorem (see also Remark 5, (i), where we compare our computations with Matsusaka estimates [K]). However this hypothesis is certainly not the best possible. It is only of the simplest form we were able to conceive. For example, we already know that, in the case of surfaces (i.e. n = 2), our Theorem holds for d > 2r 2 − 8 and i ≥ 2 when 5 ≤ r ≤ 8, and for d > 4r 2 and i ≥ r/3 when r ≥ 9 [D1]. Moreover, in the case of threefolds (i.e. n = 3), one can prove that the Theorem holds for d > 980 and i ≥ 2 when r = 7, and for d > 10r 3 and i > r when r ≥ 8. In particular we see that when n = 2 or n = 3, and d >> n, the classification of the n-dimensional smooth varieties in P2n+1 with maximal plurigenera, is complete, i.e. it holds for any i ≥ 2. When n ≥ 4, our proof of the Theorem does not allow to handle the case i = 2 (see Remark 5, (ii)). We don’t know whether the Theorem holds for any i ≥ 2 too. Recall also that Harris’ approach [H] in proving the bound P (n, r, d, 1) for the geometric genus p1 (V ) of a n-dimensional variety V , does not work in the study of plurigenera pi (V ) with i ≥ 2 ([D1], Remark 6).

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As a consequence of the Theorem, at the end of the paper we get an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef (Corollary 7). It proves that the bound presented in [D2] is, in a certain sense, a birational bound. We work over the complex number field C and we use standard notation of Algebraic Geometry (e.g. see [HT]). We will use the notation d0 (n, r) = 165n5 (r − n)3

and i0 (n, r) = 2n3 (r − n).

(0.5)

Taking into account [D1], we may assume throughout the paper n ≥ 3. We begin by showing the existence of the varieties appearing in the statement of the Theorem. Lemma 1. For any degree d > d0 (n, r), the varieties appearing in the statement of the Theorem exist, and, for any i ≥ 2, their i-th plurigenus achieves the corresponding numerical upper bound. Proof. For the existence of such varieties in V(n, r, d) we refer to [D2], Proposition (2.3). Hence we only have to compute their plurigenera. Let T ⊂ Pr be a n + 1-dimensional rational normal scroll (compare with [D2], pg.147). Denote by H (by W resp.) the hyperplane section (the n-plane of the ruling resp.) of T . Let f : F → T be the minimal rational resolution of the singularities of T . Let V ⊂ T be a smooth subvariety as in the maximal cases of the statement of the Theorem. By [D2], Lemma (2.1) and proof, we know that f induces an isomorphism f : V˜ ∼ = V,

(1.1)

where V˜ denotes the strict transform of V . We have the linear equivalence V˜ ∼ (m + 1 + a)H˜ + (
+ 1 − (a + 1)(r − n))W˜ ,

(1.2)

where a = 0 or a = 1, according to case (i) or (ii). The canonical divisor KV˜ of V˜ is linearly equivalent to the restriction to V˜ of the divisor KF + V˜ ∼ (m − n + a)H˜ + (
− 1 − a(r − n))W˜ . By [D2], Proposition (2.3) and proof, we know that if
− 1 − a(r − n) ≥ 0, then KF + V˜ is nef. This is true when
− 1 − a(r − n) < 0 too. In fact in this case
+ 1 − (a + 1)(r − n) < 0. This implies that T is smooth (otherwise, by (1.2), V˜ should contain the exceptional locus of F, in contradiction with (1.1)), and the argument developed in [D2], Proposition (2.3) and proof, applies. On the other hand, a direct computation proves that (KF + V˜ )n+1 = (m − n + a)n [(m − n + a)(r − n) + (n + 1)(
− 1 − a(r − n))] > 0. In conclusion the divisor KF + V˜ is big and nef. By Kawamata-Viehweg Vanishing Theorem, it follows that H 1 (F, OF (KF + (i − 1)(KF + V˜ )) = 0 for any i ≥ 2 (notice that this holds true also for i = 1; in fact, by [S], pg. 110, we know

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that f∗ OF = OT and R j f∗ OF = 0 for any j > 0; hence, using Leray spectral sequence and Serre Duality, we get H 1 (F, OF (KF )) = H n (F, OF ) = H n (T , OT ), and the last group vanishes because T ⊂ Pr is arithmetically Cohen-Macaulay of dimension n + 1). From the exact sequence 0 → OF (KF + (i − 1)(KF + V˜ )) → OF (i(KF + V˜ )) → OV˜ (iKV˜ )) → 0, (1.3) we deduce pi (V ) = h0 (F, OF (i(KF +V˜ ))−h0 (F, OF (KF +(i−1)(KF +V˜ )) = p(n, r, d, i, a) (see [S], pg.111).



Now we need some preliminaries (i.e. Propositions 2 and 3, and Corollary 4 below) which enable us to reduce the proof of the Theorem to an analysis of the n-folds lying on a variety of dimension n + 1 and minimal degree. Proposition 2. Let V ⊂ Pr be a projective, irreducible, smooth and nondegenerate variety of dimension n ≥ 2 and degree d. Denote by H the general hyperplane section of V and by g the linear genus of V (i.e. the genus of H n−1 ). Let E be a divisor on V . Put e = E · H n−1 . Assume e ≥ 0, and define α and β by dividing e = αd + β, 0 ≤ β < d. Put γ (n, d, g, e) = 8(n − 1)(n − 2)d 2 α(α − 1) − 16n(n − 2)αd(e + 1 − g)   +n(n − 1) (2e + d + 4)2 + (2e + d − 4g)2 . Then one has

  α+n−2 γ (n, d, g, e)/16dn(n − 1). h (V , OV (E)) ≤ n−2 0

Proof. From the exact sequence 0 → OV (E − H ) → OV (E) → OV (E) ⊗ OH → 0 we get h0 (V , OV (E)) ≤ =

∞ j1 =0 α

h0 (H, OV (E − j1 H ) ⊗ OH ) h0 (H, OV (E − j1 H ) ⊗ OH ).

j1 =0

Iterating this argument for the successive linear sections of V , we get h0 (V , OV (E)) ≤

α α−j 1 j1 =0 j2 =0

α−j1 −j2 −···−jn−3

···



h0 (H n−2 , OV

jn−2 =0

× (E − (j1 + j2 + · · · + jn−2 )H ) ⊗ OH n−2 ), where H n−2 denotes the general linear section of V of dimension 2.

(2.1)

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On the other hand, using Clifford Theorem on H n−1 , by [D1], Proposition 2, we know that, for any integer j , h0 (H n−2 , OV (E − j H ) ⊗ OH n−2 )   ≤ (2e − 2j d + d + 4)2 + (2e − 2j d + d − 4g)2 /16d. Our claim follows using this inequality and (2.1).



We need previous Proposition 2 and Castelnuovo-Halphen’s Theory ([CC], [EH]) for proving the following Proposition 3. We recall that the functions p(n, r, d, 0), d0 (n, r) and i0 (n, r) which appear in its statement, are defined in (0.4) and (0.5). Proposition 3. Let V ⊂ Pr be a projective, irreducible, smooth and nondegenerate variety of dimension n and degree d, not contained in any variety of dimension n + 1 and degree < r − n + 1. Denote by pi (V ) the i-th plurigenus of V . Assume d > d0 (n, r) and i > i0 (n, r). Then one has pi (V ) < p(n, r, d, i, 0).

(3.1)

Proof. We keep the notation of Proposition 2. We may assume pi (V ) > 0. In particular we have iKV · H n−1 = i(2g − 2 − (n − 1)d) ≥ 0. Using Proposition 2 with E = iKV , we obtain   α+n−2 0 γ (n, d, g, e)/16dn(n − 1), (3.2) pi (V ) = h (V , OV (iKV )) ≤ n−2 where g denotes the linear genus of V , and e = i(2g − 2 − (n − 1)d). By the numerical assumption on d and by the Lifting Theorem [CC], Theorem (0.2), we have that H n−1 is not contained in any surface of Pr−n+1 of degree < r − n + 1. By Castelnuovo-Halphen’s Theory [EH], Theorems 3.13 and 3.15, we get   µ g ≤ (r − n + 1) + µ(ν + 1) + 1, 2 where µ and ν are defined by dividing d −1 = µ(r −n+1)+ν, 0 ≤ ν < r −n+1. It follows that 2g − 2 − (n − 1)d ≤ d 2 /(r − n + 1). Using this bound and (3.2) we obtain       d n+1 n + 1 n−1 3 n + 1 n−2 n pi (V ) ≤ (n + 1)i i − i + (n + 1)!(r − n + 1)n 2 3 2 n +O(d ). On the other hand we have p(n, r, d, i, 0) =

  d n+1 i n+1 − (i − 1)n+1 + O(d n ). n (n + 1)!(r − n)

(3.3)

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From the two previous inequalities, it follows that  n+1 − (i − 1)n+1 i d n+1 p(n, r, d, i, 0) − pi (V ) ≥ (n + 1)! (r − n)n

n+1 n−1 3 n+1 n−2 n (n + 1)i − 2 i +2 3 i + O(d n ). − (r − n + 1)n (3.4) The coefficient of d n+1 in (3.4) is > 0 for i >> r, and so, taking into account that in the remaining term O(d n ) of (3.4) the variable i appears with exponent at most n, our claimed inequality (3.1) holds for d >> r and i >> r. A more accurate but elementary analysis of (3.4) shows that (3.1) holds for d > d0 (n, r) and i > i0 (n, r).  As a consequence of Lemma 1 and Proposition 3 we get the following Corollary 4. Corollary 4. Let V ⊂ Pr be a projective, irreducible, smooth and nondegenerate variety of dimension n and degree d. Denote by pi (V ) the i-th plurigenus of V . Assume r ≥ 2n + 1, d > d0 (n, r) and i > i0 (n, r). If pi (V ) = P (n, r, d, i) (see (0.1)) then V is contained in a n + 1-dimensional subvariety of Pr of minimal degree r − n.  Remark 5. (i) Proposition 2 should be compared with Matsusaka estimates (see [K], pg. 301–302), which give different effective upper bounds for h0 (V , OV (E)). In our context, these estimates are not as good as Proposition 2. In fact, if we apply Matsusaka estimate [K], pg. 301, (2.15.8.6) (the only one that does not require special properties for E) for E = iKV , and we use CastelnuovoHalphen bound 2g − 2 − (n − 1)d ≤ d 2 /(r − n + 1) as in the proof of Proposition 3, we obtain pi (V ) ≤

d n+1 i n + O(d n ). (r − n + 1)n

(5.1)

On the other hand, using Proposition 2 for bounding pi (V ) as in the proof of Proposition 3, we have pi (V ) ≤

d n+1 (n + 1)!(r − n + 1)n       n + 1 n−1 3 n + 1 n−2 n + O(d n ), × (n + 1)i − i + i 2 2 3

which is an upper bound more precise than (5.1). Notice also that, arguing as in Proposition 3, estimate (5.1) leads to the inequality (compare with (3.4)) p(n, r, d, i, 0) − pi (V )  n+1  d n+1 i − (i − 1)n+1 (n + 1)!i n ≥ + O(d n ), − (n + 1)! (r − n)n (r − n + 1)n

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where, contrary to (3.4), the coefficient of d n+1 is < 0 for n ≥ 3, i ≥ 2 and r ≥ 2n + 1. Therefore estimate (5.1) is not enough to deduce Proposition 3, and consequently Corollary 4. (ii) For i = 2, r ≥ 2n + 1 and n ≥ 5, the leading coefficient in (3.4) is < 0. Hence the assumption i >> r is necessary in our approach in proving Proposition 3. In the sequel we will see that the proof of the Theorem does not allow to handle the case i = 2 for n = 4 and r ≥ 9 too (compare (3.3) with (0.6) below).  We are in position to prove the Theorem. We keep the notation defined in (0.2), (0.3), (0.4) and (0.5). Fix a smooth variety V ∈ V(n, r, d). By Corollary 4 we may assume V lying on a n + 1-dimensional variety T ⊂ Pr of minimal degree r − n. Since r ≥ 2n + 1 ≥ 7, then T is a rational normal scroll (by [D2], Proposition (2.2), T cannot be a cone over a Veronese surface). First we examine the case T is a smooth rational normal scroll in Pr . In this case there exists a unique integer a = a(V , T ) such that V is linearly equivalent to (m + 1 + a)H + (
+ 1 − (a + 1)(r − n))W on T , where H (resp. W ) denotes the generic hyperplane section (resp. a n-plane of the ruling) of T (see [H], [S], [D2]). Notice that a ≥ −m. The canonical divisor KV of V is the restriction of (m − n + a)H + (
− 1 − a(r − n))W to V . We deduce that the linear genus of V is g(a) = (m + a) [(m − a − 1)(r − n) + 2
] /2, from which we get a ≤ m because g(a) ≥ 0. Therefore we have −m ≤ a ≤ m. In the range −m ≤ a ≤ − m−n−1 r−n−1 , the function g(a) is increasing and so we have   m−n−1 d 2 (r − n − 2) g(a) ≤ g − + O(d). = r −n−1 2(r − n − 1)2 Arguing as in the proof of Proposition 3, one obtains pi (V ) ≤ d n+1 i n−2 (4i 2 −2in+n2 −n)(r−n−2)n /4n!(r−n−1)2n +O(d n ). (0.6) Comparing this bound with (3.3) we deduce pi (V ) < p(n, r, d, i, 0), when d > d0 (n, r) and i > i0 (n, r). In the range m−n−1 r−n−1 ≤ a ≤ m, the function g(a) is decreasing and we have   d 2 (r − n − 2) m−n−1 = + O(d). g(a) ≤ g r −n−1 2(r − n − 1)2 Therefore, in a similar manner as before, we deduce pi (V ) < p(n, r, d, i, 0) again, when d > d0 (n, r) and i > i0 (n, r). It remains to analyze the case −

m−n−1 m−n−1 d0 (n, r), i > i0 (n, r) and − m−n−1 r−n−1 < a < m−n−1 . One has r−n−1 • if a = 0, 1, then p(n, r, d, i, a) < p(n, r, d, i, 0); • if
<
0 (n, r), then p(n, r, d, i, 1) < p(n, r, d, i, 0); • if
>
0 (n, r), then p(n, r, d, i, 0) < p(n, r, d, i, 1). Taking into account the proof of Lemma 1, one examines the remaining case T is a singular rational normal scroll, in a similar way as before, reducing the analysis on the minimal desingularization F of T . This concludes the proof of the Theorem.  Now we prove Lemma 6. To simplify the notation, put p(a) = p(n, r, d, i, a). We may write p(a) = c0 + c1 a + · · · + cn a n + cn+1 a n+1 , where cj = cj (n, r, m,
, i) are polynomials in the variables n, r, m,
and i (see (0.2) and (0.4)). First we notice that cj < 0

for any

2 ≤ j ≤ n + 1.

(6.1)

In fact we have       n n n+1 n+1 cj = i − (i − 1) (r − n) −n mn−j +1 /(n + 1)! j j −1 +O(mn−j ) (6.2) (assume O(m−1 ) = 0). Since the coefficient of mn−j +1 in (6.2) is < 0 for 2 ≤ j ≤ n + 1, and taking into account that the variable i appears in the polynomials cj with exponent at most n, inequality (6.1) holds for d >> r and i >> r. A more accurate but elementary analysis shows that (6.1) holds for d > d0 (n, r) and i > i0 (n, r). Similarly one proves that, for d > d0 (n, r) and i > i0 (n, r), one has c1 + 2c2 < 0.

(6.3)

In the range a ≥ 2, the lemma follows from (6.1) and (6.3). In order to analyze the case a ≤ −1, one first observes that, for 2 ≤ j ≤ n and a ≤ −1, one has cj + acj +1 < 0.

(6.4)

In fact, from (6.1) and the hypothesis a > − m−n−1 r−n−1 , it suffices to prove that cj −

m−n−1 cj +1 < 0. r −n−1

This follows using (6.2), taking into account that d > d0 (n, r) and i > i0 (n, r).

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By (6.4) and (6.1), we deduce that to prove Lemma 6 in the range a ≤ −1, it suffices to prove that c1 + c2 a + c3 a 2 > 0.

(6.5)

Since a > − m−n−1 r−n−1 and r ≥ 2n + 1, with similar argument as before, one sees that c1 + c2 a + c3 a 2 ≥ c1 − c2 + c3 and that c1 − c2 + c3 > 0 . This proves (6.5), and Lemma 6 for a ≤ −1. One analyzes the remaining cases in a similar manner. This concludes the proof of Lemma 6.  As a consequence of the Theorem, we get an upper bound for the self-intersection of the canonical bundle of a smooth projective variety whose canonical bundle is big and nef. To this purpose, define
 K(n, r, d) = sup KVn : V ∈ V(n, r, d) and, only for n ≥ 3, KV is nef . In [D2] we give an explicit formula for K(n, r, d), similar to the one given in the Theorem for the maximal plurigenera P (n, r, d, i). In fact we have (for r ≥ 2n + 1 and d > d0 (n, r))   if
<
0 (n, r), k(n, r, d, 0) K(n, r, d) = k(n, r, d, 1) if
>
0 (n, r),   max{k(n, r, d, 0), k(n, r, d, 1)} if
=
0 (n, r), where k(n, r, d, a) = (m−n+a)n−1 [d(m − n + a)+n(m + 1 + a)(
− 1 − a(r − n))] . Now, if M is a projective and smooth variety, birational to some V ∈ V(n, r, d), n may be bigger than K n , and K may be not nef. Nevertheless, when K is KM V M V n ≤ K(n, r, d) again. big and nef, d > d0 (n, r) and r ≥ 2n + 1, then one has KM This is the content of the following Corollary 7. It proves that, in a certain sense, K(n, r, d) is a birational bound. Corollary 7. Let M be a projective and smooth variety of dimension n ≥ 2, whose canonical bundle KM is big and nef. Assume that M is birational to a smooth and nondegenerate projective variety V ⊂ Pr of degree d > d0 (n, r), with r ≥ 2n + 1. Then n KM ≤ K(n, r, d). Proof. Using Hirzebruch-Riemann-Roch Theorem and Kawamata-Viehweg Vanishing Theorem we have n!χ (OM (iKM )) n!pi (M) n!pi (V ) = lim = lim . n n i→+∞ i→+∞ i→+∞ i i in

n KM = lim

On the other hand, by the Theorem, we deduce lim

i→+∞

n!pi (V ) n!P (n, r, d, i) ≤ lim = K(n, r, d). i→+∞ in in



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