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NUMERICAL CONSTRAINTS FOR EMBEDDED PROJECTIVE MANIFOLDS GIAN MARIO BESANA AND ALDO BIANCOFIORE Abstract. General formulas giving numerical constraints for projective invariants of embedded, complex, projective manifolds are explicitly worked out in the framework of adjunction theory.

1. Introduction The constraints imposed on a complex manifold by its being embedded in a projective space with a low codimension have been studied intensely. An nfold X can always be embedded in P2n+1 . The possibility of embedding an nfold X in P2n is related to the number of double points of a generic projection of X from P2n+1 . Double point formulas, expressing these constraints in terms of Chern classes of the manifolds and its normal bundle, can be traced back to Severi, [45], (see also Catanese, [16]). They were rediscovered and generalized by Holme, [29], and independently by Peters and Simonis, [44]. They are now essentially a special case of the Laksov-Todd double point formula, [37]. An excellent general reference is due to Kleiman, [36]. In the recent past, numerical constraints for embedded projective manifolds have been extensively utilized in the classification of complex projective varieties of given degree. This clasification was accomplished by Weil, [49], Swinnerton Dyer, [48], Ionescu, [30], [31], [33], Okonek, [43], [42], Alexander, [2], Abo, Decker and Sasakura, [1], up to degree eight. With a successful application of adjunction theory, Beltrametti, Schneider and Sommese, [8], [9], classified threefolds in P5 of degree up to twelve, while Fania and Livorni, [19], [20], classified manifolds of degree nine and ten, of dimension n ≥ 3, regardless of the codimension. Recently Bertolini, [13], classified threefolds of degree twelve in P6 . The general approach implemented by most of the above authors is a two-step process. The first step consists of taking full advantage of classical adjunction theory and endow the manifold with a special geometric structure, typically a fibration. The peculiar properties of the acquired geometric structure are then exploited in the second step in searching for constraints for the numerical characters of the manifold. 2000 Mathematics Subject Classification. Primary (14N25, 14N30, 14J) ; Secondary (14J30, 14J35, 14J40). Key words and phrases. classification, adjunction, numerical constraints. 1

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G. BESANA AND A. BIANCOFIORE

Unfortunately the second step has been conducted thus far in an ad-hoc manner, without a systematic treatment that lends itself to future uses. In this work, in the general framework of adjunction theory, we offer a systematic approach to numerical constraints for projective invariants of embedded manifolds. General relations and inequalities, obtained from Chern and Segre classes computations, are explicitly worked out for all families of special varieties arising in adjunction theory. We believe that this work will be quite useful for authors interested in similar problems. As evidence, we would like to point out that Fania and Livorni utilized several of our general formulas, while this work was in preparation, to improve the already completed classification of manifolds of degree nine and to complete their classification of manifolds of degree ten (see [20] Prop. 5.1, 6.2, 7.1 and Remark 7.5). A new application of these results to the classification of manifolds of degree eleven is given in [15]. The paper is structured as follows. In section 2 notations are fixed and results from adjunction theory are recalled for the convenience of the reader. Section 3 is devoted to the presentation of general formulas involving projective invariants of embedded manifolds. In sections 4 through 11 the above formulas are implemented according to the adjunction theoretic type of the manifold. We are greatly indebted toward an anonymous referee, whose extremely careful repeated readings of earlier versions of this work and [15] enabled us to make substantial improvements. The same referee also pointed out to the authors earlier works of M. Ohno in which some of our formulas were already obtained. Both authors would like to thank M.L. Fania for her help, patience and generosity. The authors acknowledge support from Italian MURST, through its 40 % Program Geometria Algebrica and its project Geometria sulle Variet´ a Algebriche. The first author would like to thank Helmut Epp, Dean of the School of CTI, DePaul University, for his generosity and support. The material in this paper is, in part, based upon work supported by the National Science Foundation (NSF) under Grant No. 0125068. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. 2. Background material In this section notations are fixed and definitions and results from adjunction theory, which will be used throughout the paper, are recalled. The best reference for these results is [12]. Let Vˆ be a complex projective manifold of dimension n, n-fold for short, ˆ be a very ample line bundle on Vˆ . Projective properties of (Vˆ , L) ˆ and let L N =n+m ˆ ˆ are always referred to the embedding (V , L) ⊂ P given by the comˆ plete linear system associated with L. Therefore our n-folds are always linˆ S, ˆ C, ˆ respectively a smooth threefold, early normal. We denote by X,

NUMERICAL CONSTRAINTS

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surface, curve, obtained as transverse intersection of n − 3, n − 2, n − 1, ˆ The following notation will be used throughout this general elements of |L|. work. ≈ the linear equivalence of line bundles; ≡ the numerical equivalence of line bundles; P χ(L) = (−1)i hi (L), the Euler characteristic of a line bundle L; LY or L|Y the restriction of line bundle L to a subvariety Y ; KVˆ the canonical bundle of an n-fold. When the context is clear, Vˆ may be dropped; q(Vˆ ) = h1 (OVˆ ), the irregularity of an n-fold; pg (Vˆ ) = h0 (KVˆ ), the geometric genus of an n-fold; ci (Vˆ ), ci (E), respectively the ith Chern class of an n-fold and of a vector bundle E; si (Vˆ ), the ith Segre class of Vˆ ; e(Vˆ ) = cn (Vˆ ), the topological Euler characteristic of an n-fold; ˆ n the degree of Vˆ in the embedding given by a very ample line dˆ = L ˆ bundle L; ˆ = n + dˆ − h0 (L); ˆ ∆ = ∆(Vˆ , L) ˆ κ(V ) the Kodaira dimension of Vˆ ; ˆ m = codimPN Vˆ , in the embedding given by |L|; + Si , i ∈ Z , the blow up of a surface S at i points p1 , . . . , pi . Let π : Si → S be the blow up morphism, let Ei be the exceptional divisors, and let M ∈ Pic(S). For t, r ∈ Z+ , we denote with M − tpr P the line bundle π ∗ (M ) − tj=1 rEj on Si . Let V be a 4-fold. We set R = c22 (V ); Q = c3 (V )c1 (V ); P = c2 (V )c21 (V ). Line bundles and divisors are used with little or no distinction. Additive notation will be mostly used for the group of divisors and line bundles on a variety. Multiplicative notation (juxtaposition) will be used for intersection of cycles, Chern and Segre classes. Definition 2.1. Let L be a line bundle on Vˆ . L is said to be nef if LD ≥ 0 for all effective curves D on Vˆ , and in this case L is said to be big if c1 (L)n > 0. ˆ on an n-fold Vˆ , the sectional genus Definition 2.2. For a line bundle L ˆ ˆ ˆ g = g(L) of (V , L) is defined by ˆ L ˆ n−1 . 2g − 2 = (KVˆ + (n − 1)L) 2.1. Castelnuovo’s bound and Harris’ refinement. Let Vˆ be an n-fold ˆ be a very ample line bundle on Vˆ . Let Cˆ be as above. Assume and let L ˆ ˆ = g(L) ˆ that |L| embeds Vˆ in a projective space Pn+m , n+m ≥ 4. Then g(C)

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and Castelnuovo’s Lemma (see e.g. [38, Sect. 0]) gives Ã" " #Ã # ! ! ˆ− 2 ˆ− 2 d d m ˆ ≤ (2.1) g(C) dˆ − m − 1 − −1 m m 2 where [x] means the greatest integer ≤ x. In the same notation as above, the following result due to Harris, [27], will be very useful. THEOREM 2.3 (Harris [28] Theorem 3.15). Let m ≥ 3. Set " # dˆ − 1 m1 = , ²1 = dˆ − m1 (m + 1) − 1, 1+m ½ µ1 =

1 if ²1 = m , π1 = 0 otherwise

µ

m1 2

¶ (m + 1) + m1 (²1 + 1) + µ1 .

If g > π1 and dˆ ≥ 2m + 3, then Cˆ lies on a surface of degree m. If g = π1 and dˆ ≥ 2m + 5 then Cˆ lies on a surface of degree (m + 1) or less. 2.2. Implications of the Barth-Lefschetz Theorem. Let Vˆ be an nˆ be a very ample line bundle on Vˆ and let n + m = h0 (L) ˆ − 1. If fold, let L N ∼ ∼ ˆ n − m ≥ 2, then Pic(V ) = Pic(P ) = Z. 2.3. Some special varieties. We say that a polarized pair (V, L), where L is an ample line bundle on an n-dimensional manifold V, is a scroll, or a hyperquadric fibration, or a Del Pezzo fibration over a normal polarized variety (Y, H) if there is a surjective morphism ϕ : V → Y with connected fibers such that KV + (n − dim Y + 1)L ≈ ϕ∗ (H) or, respectively, KV + (n − dim Y )L ≈ ϕ∗ (H) or KV + (n − dim Y − 1)L ≈ ϕ∗ (H). If −KV is ample we say that V is a Fano manifold. The largest positive integer a such that −KV ≈ aM for some ample line bundle M on V , is the index of V . A polarized pair (V, L) is said to be a Del Pezzo manifold if KV ≈ (1 − n)L and a Mukai manifold if KV ≈ (2 − n)L. 2.4. Reductions. Let Vˆ be a complex projective manifold of dimension n ˆ be a very ample line bundle over Vˆ . We say that the pair (V, L), and let L ˆ if L is ample and there is a morphism with V smooth, is a reduction of (Vˆ , L) π : Vˆ → V expressing Vˆ as V with a finite set F of cardinality s blown up; ˆ ≈ π ∗ L − π −1 (F ) or equivalently, K ˆ + (n − 1)L ˆ ≈ π ∗ (KV + (n − 1)L). We L V n will always write d := L . THEOREM 2.4 ([46]). Let Vˆ be a complex projective manifold of dimenˆ be a very ample line bundle on Vˆ . Let us assume that sion n and let L ˆ is nef and big. Then KVˆ + (n − 1)L ˆ and KV + (n − 1)L is ample; i) There is a reduction (V, L) of (Vˆ , L)


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ˆ > 0 for some t > 0and n ≥ 3. ii) Assume h0 (t(KVˆ + (n − 2)L)) Then KV + (n − 2)L is nef and there is a smooth surface S which is the transverse intersection of (n−2) general members of |L| such that S is a minimal model of non negative Kodaira dimension. Moreover the restriction π|Sˆ : Sˆ → S is the map into its minimal model; iii) If KV + (n − 2)L is nef and big, then S is a surface of general type. ˆ is nef and big and KV + (n − 1)L is ample, Remark 2.5. If KVˆ + (n − 1)L the pair (V, L) is unique up to isomorphism. Thus we refer to (V, L) as the ˆ reduction of (Vˆ , L). ˆ be a very THEOREM 2.6 ([47]). Let Vˆ be an n-fold with n ≥ 2 and let L ˆ is spanned by global sections ample line bundle on Vˆ . Then KVˆ + (n − 1)L unless either ˆ ∼ i) (Vˆ , L) = (Pn , OPn (1)) or (P2 , OP2 (2)); ˆ ∼ ii) (Vˆ , L) = (Qn , OQ (1)) where Qn is a smooth quadric in Pn+1 ; ˆ to iii) Vˆ is a Pn−1 -bundle over a smooth curve and the restriction of L a fiber is OPn−1 (1). ˆ is spanned by global sections. The morSuppose that KVˆ + (n − 1)L phism given by the complete linear system associated with this line bundle, Φ : Vˆ → PN , is called the adjunction map. We will write Φ = σ ◦ r for the Remmert-Stein factorization of Φ, so r : Vˆ → Y is a morphism with connected fibers onto a normal variety Y and σ is a finite map. ˆ be an n-fold and let L ˆ be a very ample THEOREM 2.7 ([47]). Let (Vˆ , L) ˆ is spanned by global sections. line bundle on Vˆ . Assume that KVˆ + (n − 1)L Then there are the following possibilities: ˆ ; i) dim Φ(Vˆ ) = 0 and KVˆ ≈ −(n − 1)L ii) dim Φ(Vˆ ) = 1 and the general fiber of r is a smooth quadric Q with ˆ | ≈ OQ (1); L Q iii) dim Φ(Vˆ ) = 2 < n, r is a Pn−2 -bundle over a smooth surface Y and ˆ to a fiber is OPn−2 (1); the restriction of L ˆ iv) dim Φ(V ) = n. THEOREM 2.8 ([47]). With the same notation as in Theorem 2.7, if the adjunction map has n-dimensional image, then r : Vˆ → Y expresses Vˆ as the blowing up of a finite set F on a smooth projective manifold V = Y. If ˆ L = OV (r(L)), then L and KV + (n − 1)L are ample, ∗ ˆ and (V, L) is the reduction of (Vˆ , L). ˆ r (KV + (n − 1)L) ≈ KVˆ + (n − 1)L THEOREM 2.9 ([46] , [22]). Let Vˆ be a complex projective manifold of ˆ be a very ample line bundle over Vˆ and let (V, L) be dimension n ≥ 3, let L ˆ ˆ a reduction of (V , L). Then KV + (n − 2)L is nef and big unless either: i) (V, L) ∼ = (P4 , OP4 (2)) or (P3 , OP3 (3));

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ii) (V, L) ∼ = (Q, OQ (2)), where Q is a hyperquadric in P4 ; iii) there is a surjective morphism ϕ : V → Y onto a smooth curve Y , whose general fiber is (P2 , OP2 (2)) and 2KV + 3L ≈ ϕ∗ H for some ample line bundle H on Y ; iv) (V, L) is a Mukai manifold; v) (V, L) is a Del Pezzo fibration over a smooth polarized curve (Y, H); vi) n ≥ 4 and (V, L) is a scroll over a polarized threefold (Y, H); vii) (V, L) is a hyperquadric fibration over a polarized surface (Y, H). In the cases v) through vii) of the above theorem we denote by ϕ : V → Y the map which coincides with the Remmert Stein factorization of the morphism associated to a sufficiently high power of KV + (n − 2)L. Therefore in all these cases KV + (n − 2)L ≈ ϕ∗ (H) for an ample line bundle H on Y. Note that by [11] (3.1), V = Vˆ in case vi) and also in case vii), if n ≥ 4. Note that in case iii), by [21] and [11], every fiber is P2 . ˆ If KV + (n − 2)L is Definition 2.10. Let (V, L) be the reduction of (Vˆ , L). nef and big then (V, L) is said to be of log-general type. ˆ be a very THEOREM 2.11 ([9](4.4.1)). Let Vˆ be an n-fold, and let L ˆ − 1. Let (V, L) ample line bundle on Vˆ . Let n > m, where n + m = h0 (L) n ˆ ˆ ˆ ˆ be the reduction of (V , L). If d 6= 2 − 1, then V = V. ˆ 2.5. Pluridegrees. Let (Vˆ , L)be an n-fold admitting (V, L) as its reducˆ ˆ ˆ ˆ tion. Let (X, LXˆ ), (S, LSˆ ) be its general threefold and surface section and let (X, LX ) and (S, LS ) be their reductions. Following [6] we define: ˆ jL ˆ n−j ; dˆj = (KVˆ + (n − 2)L) dj = (KV + (n − 2)L)j Ln−j . Notice that dˆ0 = dˆ and d0 = d; d1 = KS LS ; d2 = KS2 ; d3 = (KX + LX )3 . If n = 4, then d4 = (KV + 2L)4 . Notice also that because 2g − 2 = d + d1 , d and d1 must be both odd or both even.


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3. Chern and Segre Classes Formulas In this section several formulas involving the Chern and Segre classes of varieties, line bundles and their Jet bundles are presented. In particular, the basic double point formulas are recalled. Double point formulas can be traced back to Severi, [45], and were rediscovered by Holme [29] and Peters and Simonis, [44]. An excellent general reference is due to Kleiman, [36]. Segre classes are often used for the particularly nice compact forms that they allow. 3.1. Double Point Formulas. Lemma 3.1 (compare [36], [38], [5], [6], [10]). Let Vˆ be an n-fold in Pn+m . ˆ = OPn+m (1) . Then Let L |Vˆ ¶ n µ X 2n + 1 2 ˆ ˆk ≥ 0 d − (3.1) sn−k (Vˆ )L k k=0

with equality if n ≥ m. If m + 1 = n then (3.2)

ˆ n−1 = dˆL

n−1 Xµ k=0

2n k

¶ ˆk. sn−k−1 (Vˆ )L

Lemma 3.2 (compare [38],[6]). Let Vˆ be a smooth n-fold in Pn+m . Let ˆ = OPn+m (1) . Let X ˆ be a general threefold section and let Sˆ be a general L |Vˆ surface section. Then ¶ n µ X 2n ˆk ≥ 0 (3.3) sn−k (Vˆ )L k k=0

with equality if n ≥ m + 1; ˆ ≥ e(X) ˆ − 2g + 2. 2e(S)

(3.4)

ˆ be an n-fold admitting (V, L) as its reduction, as Lemma 3.3. Let (Vˆ , L) in (2.4). Then ¶ n µ X 2n + 1 2 (3.5) sn−k (V )Lk + (2(n+1) − 1)s ≥ 0 (d − s) − k k=0

with equality if n ≥ m;

(3.6)

¶ n µ X 2n sn−k (V )Lk − 2n s ≥ 0 k k=0

with equality if n ≥ m + 1. Proof. A simple application of [26, p. 298], gives these translations of the above (3.1) and (3.3). ¤

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ˆ be an n-fold admitting (V, L) as its 3.2. Schur polynomials. Let (Vˆ , L) reduction, as in (2.4). Recall (see for example [6]) that the first holomorphic ˆ is a rank n + 1 vector bundle on Vˆ , which is generated by jet bundle J1 (L) its global sections, while J1 (L) is a rank n + 1 vector bundle on V, which is generated by its global sections outside of a finite set. Following [26, p. 216], let λ = (λ1 , . . . , λt ) be a partition of a positive P integer t, i.e. t1 λi = t, with 0 ≤ λt ≤ · · · ≤ λ1 ≤ n + 1. ˆ = det(cλ +j−i (J1 (L))) ˆ Let ∆λ (J1 (L)) for 1 ≤ i, j ≤ t, be the Schur polynoi ˆ mial associated with λ, in the Chern classes of J1 (L). It follows from [26, p. 216] that, for all effective k-cycles Yk ⊂ Vˆ , with k ≥ t, it is ˆ k ≥ 0. (3.7) ∆λ (J1 (L))Y ˆ L ˆ ˆ ) be a smooth threefold, admitting (X, LX ) as its Lemma 3.4. Let (X, X ˆ L ˆ ˆ ) and (S, LS ) be the general surface section reduction, as in (2.4). Let (S, S and its reduction. Then 1) e(X) ≤ 24χ(OS ) + d + d1 − 2d2 ; 2) e(X) ≥ 24χ(OX ) − 24χ(OS ) − 8d − 7d1 − d2 ; 3) e(X) ≤ 48χ(OX ) − 72χ(OS ) + 10d + 12d1 + 9d2 + d3 − 8s. Proof. The Chern and Segre classes of J1 (LX ) can be computed in terms ˆ ˆ )) = c3 (J1 (LX )), of the Chern and Segre classes of X. Notice that c3 (J1 (L X ˆ ˆ )) = s3 (J1 (LX ))+8s. ˆ ˆ )) = (c1 c2 −c3 )(J1 (LX )) and s3 (J1 (L (c1 c2 −c3 )(J1 (L X X Now compute the Schur polynomials corresponding to partitions of t = 3 ˆ and use (3.7) with Y3 = X. ¤ ˆ L ˆ ˆ ) be a smooth threefold, admitting (X, LX ) as its Lemma 3.5. Let (X, X ˆ L ˆ ˆ ) and (S, LS ) be the surface section and its reduction, as in (2.4). Let (S, S reduction. Assume that, for some rational number a ∈ Q, a tensor power of Da = KX + aLX is effective. Then 1) 3(a − 1)d + 2ad1 − (a − 3)d2 + 12aχ(OS ) − 24χ(OX ) ≥ 0; 2) 6(a − 1)d + (4a + 3)d1 + 2(a + 1)d2 + d3 − 12aχ(OS ) + 24χ(OX ) ≥ 0. Proof. The Chern and Segre classes of J1 (LX ) can be computed in terms of the Chern and Segre classes of X. Let π be the reduction morphism, as in ˆ ˆ ))π ∗ (Da ) = c2 (J1 (LX ))Da and (2.4).Notice that c2 (J1 (L X 2 ∗ ˆ ˆ ))π (Da ) = (c2 − c2 )(J1 (LX ))Da . (c1 − c2 )(J1 (L 1 X Now compute the Schur polynomials corresponding to partitions of t = 2 and use (3.7) with Y2 = π ∗ (Da ). ¤ 4. Scrolls over Curves In this section we collect general results on scrolls over curves. In particular, double point formulas are implemented for this class of manifolds. The following Lemma was already essentially contained in [40, Lemma 3.1 and 3.3] and [41, Lemma 4.1 and 4.2] .


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ˆ ⊂ Pn+m be an n-dimensional scroll over a curve Lemma 4.1. Let (Vˆ , L) Y of genus q. Then dˆ2 ≥ (2n + 1)dˆ + n(n + 1)(q − 1) with equality if n ≥ m; dˆ ≥ n(1 − q) with equality if n ≥ m + 1. Proof. Recalling that Vˆ = P(E) for a vector bundle E over Y , standard Chern classes computation using the formula ˇ ⊗ OP(E) (1)) ct (P(E)) = ρ∗ (ct (Y ))ct (ρ∗ (E) ˆ q, L ˆ i and allow us to compute the ith Segre class si (Vˆ ) = si in terms of d, i−1 ˆ F where F is a generic fiber of the natural projection ρ : P(E) → Y. It L is (4.1) (−)i

h³

n+i−1 i

´

î − L

³³

si = ´ ³ (2q − 2) +

n+i−2 i−1

n+i−1 i−1

Combine these with (3.1) and (3.3) to get the result.

´ ´ i ˆ i−1 F . dˆ L ¤

The following Lemma is due to Ionescu. A proof can essentially be found in [35, Corollary 1] and [33, Proposition (5.2)]. ˆ = (P(E), OP(E) (1)) be an n-dimensional smooth Lemma 4.2. Let (Vˆ , L) ˆ very ample, over a hyperelliptic curve C. Then h1 (E) = 0, scroll with L ˆ = ng and there exists such a scroll if and only if ∆(Vˆ , L) i) dˆ ≥ n(g + 1) + 2 if g ≥ 2; ii) dˆ ≥ 2n + 1 if g = 1. A simple application of [35] gives the following Lemma. A proof is contained in [3]. ˆ be an n-dimensional scroll of degree dˆ over a smooth Lemma 4.3. Let (Vˆ , L) ˆ ≥ 2n + g − 3. curve of genus g ≥ 2. If dˆ > 2g − 2 then ∆(Vˆ , L) The following Lemma is also inspired by Ionescu, [30]. We thank an anonymous referee for the shortened proof. ˆ be an n-dimensional scroll, n ≥ 2 of degree dˆ over Lemma 4.4. Let (Vˆ , L) ˆ ≥ g + 1. a curve of genus g ≥ 1 with dˆ ≥ 2g + 1. Then ∆(Vˆ , L) ˆ be a scroll with dˆ ≥ 2g + 1 and ∆ ≤ g. Proof. By contradiction, let (Vˆ , L) Then, by [23, Theorem (3.5) 3)] , it is h1 (Vˆ , OVˆ ) = 0. This contradicts the assumption g ≥ 1. ¤

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5. Scrolls over Surfaces In this section we collect general results on scrolls over surfaces. We begin with a remark that will allow us to easily rewrite the double point formulas for these varieties. ˆ and Remark 5.1. In case iii) of Theorem 2.7, Vˆ ∼ = P(E), where E = r∗ (L) ˆ is the tautological line bundle on P(E). It is known, see for example [12, L Section 11.1] , that Sˆ is a meromorphic (non-holomorphic) section of r and ˆ L ˆ | ). Thus the restricted morphism r| is just the reduction morphism of (S, ˆ S

ˆ S

ˆ L ˆ | ), where L = detE. Notice that Sˆ is then Y (Y, L) is the reduction of (S, ˆ S blown up at c2 (E) points and χ(OY ) = χ(OSˆ ). The following Lemma was already contained in [40, Lemma 3.3] and [41, Lemma 4.1 and 4.2] . Lemma 5.2. Let E be a vector bundle of rank n − 1 over a surface Y and ˆ = (P(E), OP(E) (1)). Assume Vˆ ⊂ Pn+m with L ˆ = OPn+m (1) . Let let (Vˆ , L) |Vˆ ˆ S be a general surface section. Then (5.1)

ˆ 12(dˆ2 − d) ≥ 4(4 − n)dˆ + 4(2n + 1)(g − 1)+ 2 n +n (n2 + n − 2)(KS2ˆ − 6χ(OSˆ )) + (n − 1)(n − 2)c2 (E) with equality if m ≤ n;

(5.2)

(3 − n)dˆ + 2n(g − 1) − 2(n2 − 1)χ(OSˆ )+ 1/3(n2 − 1)KS2ˆ + 1/3(n − 1)(n − 2)c2 (E) ≥ 0 with equality if m ≤ n − 1.

Lemma 5.3. Let E be a very ample vector bundle of rank 2 over a surface ˆ L ˆ ˆ ) = (P(E), OP(E) (1)). Assume dˆ ≥ 10 and dˆ 6= 13. Then Y and let (X, X c2 (E) ≥ 4. Proof. By the above Remark 5.1 it is c1 (E)2 = dˆ + c2 (E). Ampleness of E, by [4] gives c1 (E)2 ≤ (c2 (E) + 1)2 which implies dˆ ≤ c2 (E)2 + c2 (E) + 1. Therefore if dˆ ≥ 8 it is c2 (E) ≥ 3. Let now c2 (E) = 3. If dˆ ≥ 10, by [39], it follows that (Y, E) = (P2 , OP2 (1) ⊕ OP2 (3)). This gives dˆ = 13. ¤ 6. Quadric Fibrations over Curves In this section numerical constraints for hyperquadric fibrations over a curve are presented. ˆ and r be as in Theorem 2.7 case ii), i.e (Vˆ , L) ˆ is a hyperquadric Let (Vˆ , L) ˆ fibration over a smooth curve Y of genus q = g(Y ). As L is very ample, [30,


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Lemma 0.6] shows that if n ≥ 3 all the fibers of the map r are irreducible and reduced. Therefore by [21, (3.1) - (3.7)] , there is a vector bundle E of ˆ = ι∗ (OP(E) (1)). Note rank n+1 and an embedding ι : Vˆ → P(E), such that L that Vˆ is a divisor on P(E) and Vˆ ∈ |OP(E) (2) + ρ∗ B| for some B ∈ Pic(Y ), where ρ : P(E) → Y is the projection map. Let b = deg B, ² = deg E, T = c1 (OP(E) (1)), F a generic fiber of ρ. Then (6.1)  ˆ  d = 2² + b b + ² + 2q = g + 1   2² + (n + 1)b ≥ 0

 ˆ  b = −d − 4q + 2g + 2 or equivalently ² = dˆ + 2q − g − 1   ˆ + dˆ ≥ 0 n(2 + 2g − 4q − d)

By [32, p.467] there are no exceptional divisors E on Vˆ such that Ê ) ∼ (E, L = (Pn−1 , OPn−1 (1)) and [E]E ∼ = OPn−1 (−1), if n ≥ 3. Furthermore if q(Y ) = 1, and E is ample, by [21] (3.8),(3.11) ˆ = h0 (Y, E) = ² h0 (Vˆ , L)

(6.2)

The following Lemma, in the same spirit of Lemma 5.2, will be very useful. It was already essentially contained in [41, Lemma 4.1 and 4.3], and [40, Lemma 3.4] . ˆ be an n-dimensional hyperquadric fibration over a Lemma 6.1. Let (Vˆ , L) ˆ = OPn+m (1) . Let ², b, q, g, dˆ be as smooth curve Y , where Vˆ ⊂ Pn+m and L |Vˆ in (6.1). Then (6.3)

ˆ dˆ − 4n + 1) d( − g + 1 − 2(n2 − 1)(q − 1) ≥ 0 2

with equality if n ≥ m; (6.4)

dˆ + (2n − 1)(q − 1) ≥ 0

with equality if n ≥ m + 1. Remark 6.2 (compare [34],[40] Lemma 3.4). If n ≥ m + 1 then (6.4) gives ˆ = ² + n + 1 and so ² = m. q = 0 and dˆ = 2n − 1. If q = 0 then h0 (L) The following lemma is essentially due to Fujita who pointed it out to ˆ is very ample. Fania and Livorni. Recall that here E is spanned as L ˆ be a Lemma 6.3 (Fujita, cf [20] Proof of Proposition (5.1)). Let (Vˆ , L) 1 hyperquadric fibration over P with ² ≤ n. Then b ≤ 1. ˆ be spanned. An easy argument shows Remark 6.4. Let KVˆ + (n − 1)L ˆ ˆ ˆ is a that if (S, L|Sˆ ) is a conic bundle over a smooth curve Y, then (Vˆ , L) hyperquadric fibration over Y . Of course the converse is also true.

12


7. Results on blow-ups of (P3 , OP3 (3)), (P4 , OP4 (2)) and (Q, OQ (2)) ˆ as in case i) In this section the double point formulas for varieties (Vˆ , L) and ii) of Theorem 2.9 are presented. ˆ be an n-fold in Pn+m admitting a reduction (V, L). Lemma 7.1. Let (Vˆ , L) 1) If (V, L) = (P3 , OP3 (3)), then s ≤ 14

and

dˆ ≥ 13;

2) If (V, L) = (P4 , OP4 (2)), then s≤1

and

dˆ ≥ 15;

3) If (V, L) = (Q, OQ (2)), then s≤7

and

dˆ ≥ 9.

ˆ it follows from (3.5) Proof. In case 1), computing the Segre classes of (Vˆ , L), that (s−25)(s−14) ≥ 0. On the other hand it follows from (3.6) that s ≤ 20, so that s ≤ 14. Because dˆ = 27 − s it is dˆ ≥ 13. In case 2), blowing up a point we obtain (P(E), OP(E) (1)) where E = OP3 (1) ⊕ OP3 (2). Using successive hyperplane sections we can argue on the surface case and see easily that by blowing up another point we will certainly loose the very ampleness of Vˆ . Therefore s ≤ 1 and thus dˆ ≥ 15. In case 3) one can proceed exactly as in case 1) and see that s ≤ 7 and thus dˆ ≥ 9. ¤ 8. Manifolds whose reduction is a P2 -bundle over a curve. ˆ be as in Theorem 2.9, case iii). Then (V, L) is a P2 -bundle Let (Vˆ , L) over Y , see [12]. Therefore there exists a vector bundle E of rank 3, on Y, such that V = P(E). Let ρ : P(E) → Y be the projection, T = c1 (OP(E) (1)) and F be a generic fiber of ρ. Let ² = T 3 . It is L = 2T + ρ∗ (B) where B is a line bundle of degree b on Y. ˆ ∈ Pn+m be as in Section 8. Then: Lemma 8.1. Let (Vˆ , L) 5s 35dˆ − 96(q − 1) − ≥0 dˆ2 − 2 2 with equality if m ≤ 3.

(8.1)

Proof. Because V is a projectivized bundle, we can compute its Segre classes: (8.2) (8.3) (8.4) Now use (3.5).

s1 (V )L2 = 8(q − 1) − d; d s2 (V )L = −12(q − 1) + ; 2 s3 (V ) = 12(q − 1). ¤


13

9. Manifolds whose reduction is a Mukai manifold ˆ be an n-fold in Pn+m , admitting a reduction (V, L) Lemma 9.1. Let (Vˆ , L) ˆ L ˆ ˆ) as in case iv) of Theorem 2.9, i.e. which is a Mukai manifold. Let (X, X be a general threefold section, and (X, LX ) its reduction. Then (9.1)

5s − 120 − dˆ2 + 20dˆ ≤ e(X) ≤ 10dˆ − 96 + 2s

with equality on the left hand side if m ≤ 3. Proof. Using the fact that KX = −LX the Segre classes of X can be computed. (9.2)

s3 (X) = −e(X) + 48 − d;

(9.3)

s2 (X)LX = −24 + d;

(9.4)

s1 (X)L2X = −d.

Now use (3.5) for the left hand side and (3.6) for the right hand side.

¤

10. Manifolds whose reduction is a Del Pezzo Fibration ˆ is a smooth n-fold, n ≥ 3, embedded in Pn+m , In this section (Vˆ , L) admitting a reduction (V, L) as in case v) of Theorem 2.9, i.e. a Del Pezzo fibration ϕ : V → Y over a curve Y with KV + (n − 2)L ≈ ϕ∗ (H) for an ample line bundle H on Y. Let F be a generic fiber of ϕ, we set f = Ln−1 |F . It ˆ ˆ is 3 ≤ f ≤ 9. If (X, LXˆ ) is a general threefold section, it admits a reduction (X, LX ) which is also a Del Pezzo Fibration over Y with KX + LX ≈ ϕ∗ (H). ˆ L ˆ ˆ ) be a general surface section and let (S, LS ) be its reduction. Let Let (S, S pg = pg (S), q = q(S). The following facts are known, see [17] and [11], or easy to show : (10.1)

deg(H) = pg + q − 1

q = g(Y )

χ(OX ) = 1 − q

It is also known that S is a minimal model with κ(S) = 1 and d2 = 0. Therefore it is χ(OS ) ≥ 0 and thus (10.1) gives pg ≥ 1. Notice that we have 2g − 2 − d = d1 = (pg + q − 1)f ; KX L2X = 2(g − 1) − 2d; (10.2)

2 KX LX = −4(g − 1) + 3d; 3 KX = 6(g − 1) − 4d;

c2 (X)LX = 12χ(OS ) − 2(g − 1) + d. Lemma 10.1. In the notation of 10, it is : (10.3)

e(X) ≥ −36(1 − q) − 84pg + 34(g − 1) + 3dˆ − dˆ2 − 12s

with equality when m ≤ 3;

14


60(1 − q) + 108pg − 32(g − 1) − 3dˆ + dˆ2 + 12s ≥ 0; s (5dˆ − dˆ2 ) 5(g − 1) (10.5) pg ≥ (q − 1) − + + . 6 12 6 Proof. Use (3.1), (3.4), and the classical double point formula for surfaces. ¤

(10.4)

The global structure of 3-dimensional Del Pezzo fibrations was studied by D’Souza and Fania[17],[18] and by Fujita [24]. Their results can be used to obtain numerical constraints on the invariants of these varieties. ˆ be as in 10. Assume f = 3. Lemma 10.2. Let (Vˆ , L) ˆ dˆ − 15) d( (10.6) If n ≥ 3 then − g − 33q + 34 ≥ 0 2 and equality holds if m ≤ 3. ˆ dˆ − 21) d( − g − 63q + 64 ≥ 0 (10.7) If n ≥ 4 then 2 and equality holds if m ≤ 4. ˆ L ˆ ˆ ) be a general threefold section and let (X, LX ) be its reProof. Let (X, X ˆ L ˆ ˆ ) = (X, LX ). Following D’Souza [17], duction. According to [11], it is (X, X ˆ which turns out to be a ˆ L ˆ ˆ ) can be embedded as a divisor in P(ϕ∗ (L)), (X, X 3 P -bundle over Y. If T denotes a divisor in the linear system of the tautologˆ it is X ˆ = 3T + ρ∗ (B) for some line bundle B on ical line bundle of P(ϕ∗ (L)), ˆ Y, where ρ : P(ϕ∗ (L)) → Y is the natural projection. Setting b = degB, and letting F denote the numerical class of a generic fiber of the P3 -bundle, it is ˆ ≡ 3T + bF. Standard Chern classes computations and (3.1) give (10.6). X ˆ L ˆ ˆ ) be the fourfold section. D’Souza’s construction can be If n ≥ 4, let (X, X reproduced in this case by using ϕ∗ (L). The same argument as above gives (10.7). ¤ Del Pezzo fibrations whose reduction (X, LX ) is a blow up of a P2 -bundle over Y , along 9 − f disjoint sections σi (Y ), were considered by D’Souza and Fania, [18]. Following their work, numerical constraints can be achieved for this class of threefolds. ˆ be as in 10, embedded in Pn+m . Let (X, ˆ L ˆ ˆ ) be Lemma 10.3. Let (Vˆ , L) X a general threefold section with reduction (X, LX ). Assume X is the blow up π : X → P(E), along t = 9 − f disjoint sections where ρ : P(E) → Y is the projectivization of a rank three vector bundle over Y . Assume M ≈ OP(E) (3) + ρ∗ (B) for some line bundle B on Y, so that LX ≈ π ∗ (M ) − Z where Z is the line bundle associated with the exceptional divisor. Then (10.8) dˆ2 − 20dˆ ≥ (24 − 32f )(1 − q) − (84 − 17f )χ(OS ) + 5s with equality if m ≤ 3.


15

Proof. Let ² = deg c1 (E). Notice that deg H = 2q −2+²+b where b = deg B and therefore, recalling (10.1), χ(OS ) = ² + b. Let Ni be the normal bundles of the sections σi (Y ) in P(E) and let ri = deg c1 (detNi ). By means of [26, ˆ and c1 (X) ˆ 3 can be expressed in terms of the Chern classes of p.300] c3 (X) P(E) , the ri ’s and q. Now use (3.1) to get the result. ¤ When f = 4, following Fujita [24], one sees that (V, L) can be embedded in P(E) where E = ϕ∗ (−KV ) so that OP(E) (1)| = L. This information V allows us to find further numerical constraints in this case. ˆ (V, L), be as in 10. Assume f = 4. Then Lemma 10.4. Let (Vˆ , L), (10.9)

dˆ2 − 8(n − 1)dˆ − 8(2n2 − 2n − 1)(q − 1) −4(g − 1) + (2n+1 − 8n + 7)s ≥ 0

with equality if m ≤ n. Proof. Following Fujita, [24], let E = ϕ∗ (−KV ). Notice that this is a rank n + 2 vector bundle over Y and let ρ : P(E) → Y be the natural projection. Let R = ρ∗ (OP(E) (2) ⊗ IV ). Then the conormal bundle of V in P(E) is N ∗ = ϕ∗ (R) ⊗ OP(E) (−2). Standard Chern classes computations, along the lines of Lemma 4.1, give the following equality : (10.10) sk (V )Ln−k = (−1)k [4(Ak − 4Ak−1 + 4Ak−2 ) − 4(Ck − 4Ck−1 + 4Ck−2 )] ² +(−1)k [8(Bk − 4Bk−1 + 4Bk−2 )(1 − q) + 2(Ak − 6Ak−1 + 8Ak−2 )b] where

µ

¶ n+1+k Ak = ; k µ ¶ n+k Bk = ; k−1 µ ¶ n+1+k Ck = ; k−1 ² = deg c1 (E); b = − deg c1 (R).

Notice that d = 4² + 2b and (g − 1) = 4(q − 1) + b + d. Now plug the above equalities into (3.5). ¤ If n ≥ 4 the generic fiber F of the fibration ϕ : V → Y is a Del Pezzo ˆ | ≈ L| . manifold of dimension n − 1 with KF ≈ −(n − 2)L|F . Notice that L F F ˆ | and using the double point Considering the embedding of F given by L F formula we can obtain a bound on the Euler characteristic of F, in the case n = 4.

16


ˆ be as in 10, n ≥ 4. Let (W ˆ ,L ˆ | ) be the fourfold Lemma 10.5. Let (Vˆ , L) ˆ W section and let F be a generic fiber of the Del Pezzo fibration of the reduction ˆ over Y. Then W of W (10.11)

(13 − f )f − 36 ≤ e(F ) ≤ 6f − 24

Proof. Because KF ≈ −2LF , Kodaira vanishing gives χ(OF ) = 1. The Segre classes of F can then be computed: (10.12)

s3 (F ) = −e(F ) + 48 − 8f ;

(10.13)

s2 (F )LF = 4f − 12;

(10.14)

s1 (F )L2F = −2f.

Using (3.1) and (3.3) the result is obtained.

¤

Lemma 10.6. In the same notation of the above Lemma 10.5 assume m = 3. Then ˆ − 48 e(F ) = (24 − d)f Proof. Restricting (3.2) to F we have: ˆ 3F = s3 (F ) + 8s2 (F )L ˆ F + 28s1 (F )L ˆ 2F + 56L ˆ 3F . dˆL Using the Segre classes computed in Lemma 10.5 the result is obtained. ¤ 11. Manifolds whose reduction is a hyperquadric fibration over a surface ˆ be an n-fold in Pn+m admitting a reduction In this section let (Vˆ , L) (V, L) as in case vii) of Theorem 2.9, i.e. a quadric fibration over a surface ˆ L ˆ ˆ ) be a general threefold section and let (X, LX ) be its (Y, H). Let (X, X ˆ L ˆ ˆ ) be a general surface section and let (S, LS ) be its reduction. Let (S, S reduction. Throughout this section, for simplicity, the following notation is fixed: h0 := h0 (KX + LX ). Notice that in the hypothesis of 11, S is a minimal surface of general type and it is: (KX + LX )3 = 0; (11.1)

χ(OX ) = χ(OS ) − h0 ; h1 (OX ) = h1 (OS ) = h1 (OY ) = q; d2 = 2H 2 ≥ 2; c2 (X)L = e(S) − 2(g − 1) + d.

For general results on quadric fibrations we refer the reader to [14] and [5]. Please note that Theorem (2.2) in [5] is stated incorrectly, as the failure of h0 (KX + LX ) ≥ 2 implies pg (S) = q(S) = 1 or 2. Most of the results that are needed are summarized below : THEOREM 11.1 ([14], [5]). In the notation and hypothesis of 11 : i) Y is smooth, H = KY + L for some ample line bundle L on Y ;


17

ii) If L2 ≥ 5 then H and KV + (n − 2)L are spanned; iii) h0 (KV + (n − 2)L) ≥ 2 unless h0 = 1, pg (S) = q = 1, 2 and (Y, L) is a minimal surface of genus g(Y, L) = 2. If h0 = 1 and q = 2, then d2 = 4. Lemma 11.2. In the notation and hypothesis of 11, it is : ˆ − d) ˆ − 48h0 + 34(g − 1) − 3e(S) + 8d2 − 12s (11.2) e(X) ≥ d(3 with equality if m ≤ 3. Proof. This is just a translation of (3.5) in this case.

¤

Lemma 11.3. In the notation and hypothesis of 11 it is: (11.3)

e(X) − 2e(S) ≤ 2q − 2 − 4h0 .

Proof. The argument in the proof of [5, Theorem (1.3)] can be carried on in general giving the above inequality. ¤ In the following Propositions, the classification of smooth polarized surfaces of sectional genus g = 2 plays a fundamental role. Such a classification is essentially due to Beltrametti, Lanteri and Palleschi, [7], and Fujita, [21], with recent developments concerning the existence of some of the surfaces due to Fujita and Yokoyama, [25]. A nice synthesis is found in [23], which will be used as main reference. Proposition 11.4. In the notation and hypothesis of 11, if h0 = 2 and d2 ≤ 6 then (Y, L) is listed in the following table with its invariants: (11.4) # d2 pg q g κ (Y, L) 2 1 2 0 0 2 −∞ (P8 , OP2 (6) − 8p2 ) 2 2 4 0 0 2 −∞ (P10 , OP2 (9) − 8p3 − 2p2 ) 3 4 0 0 2 0 minimal Enriques surface, L2 = 2 4 6 0 0 2 1 minimal elliptic surface L2 = 1 5 6 0 1 3 −∞ (P(E)1 , OP(E) (4) + ρ∗ (D) − p2 ) e = 0, −1 where ρ : E → C is a rk 2 vector bundle, g(C) = 1 deg D = 2e + 1 Proof. Because h0 = 2 the above Theorem 11.1 gives (11.5)

L2 ≤ 4.

Recalling that S is a minimal surface of general type, it is pg (S) ≤ Using (11.1) we get d2 (11.6) pg (Y ) ≤ . 2 Riemann Roch theorem for KY + L and (11.1) give

(11.7)

L(KY + L) = 2(1 − pg (Y ) + q), d2 KY (KY + L) = − 2(1 − pg (Y ) + q). 2

d2 2

+ 2.

18


Two separate cases will now be considered according to the Kodaira dimension of Y. Case 1. κ(Y ) ≥ 0. In this case the fact that χ(OY ) ≥ 0, (11.7), ampleness of L and of H = KY + L, give q = pg (Y ), d2 ≥ 4, g(Y, L) = 2, g(Y, H) = d22 and L2 = 1, 2. If L2 = 1, the classification found in [23] (15.2) and (15.9) gives case 4 in the statement. If L2 = 2 then KY L = 0 and thus KY is numerically trivial. The classification in [23] (15.2) and (15.7) gives case 3. Case 2. κ(Y ) < 0. In this case pg (Y ) = 0 and (11.1) gives pg (S) = 2. Because S is of general type, χ(OS ) ≥ 1 and thus q ≤ 2. From (11.7) one can see that Y cannot be P2 and thus KY +2H is nef (see e.g. [23] (11.2)). Therefore (KY +2H)2 ≥ 0. Assume Y is a P1 -bundle over a curve. Then KY2 = 8(1 − q), and thus it follows from (11.7) that (KY + 2H)2 = (KY + 2(KY + L))2 = 4d2 − 16q ≥ 0. Recalling that d2 ≤ 6 it follows that q ≤ 1. Then again from (11.7) it follows that L2 = 4 + 4q − d22 + KY2 = 12 − 4q − d22 ≥ 5 which contradicts (11.5). Therefore, since Y is neither P2 nor a P1 -bundle, it must be KY + H nef (see e.g. [23] (11.7)). Therefore (2KY + L)2 ≥ 0. Combining the last inequality with (11.7), the following possible cases are left:

(11.8)

q d2 L2 g(Y, L) H 2 g(Y, H) 0 2 4 2 1 1 0 4 ≤4 2 2 2 0 6 ≤4 2 3 3 1 6 4 3 3 2

A simple check using [23] (15.2) and (12.3) gives the remaining cases in the statement. ¤ Proposition 11.5. In the notation and hypothesis of 11, assume h0 ≥ 3 and d2 ≤ 6. Then κ(Y ) ≤ 0 and the possible invariants of (Y, H) are listed


19

in the following table: # d2 h0 pg (Y ) 1 2 3 0 2 4 3 1 3 4 3 0 4 4 4 0 5 6 3 0

q 0 0 0 0 0

6

6

3

0

1

7 8 9

6 6 6

3 4 5

0 0 0

0 0 0

(Y, H) P2 (1)) Y is a K3 surface H ' L (P27 , OP2 (3) − 7p) (Q, OQ (1)) (Fe,9 , 2C0 + (3 + e)f − 9p) e = 0, 1, 2 Y = P(E), E rk 2 vector bundle over C c1 (E) = 1, g(C) = 1 C0 ∈ |OP(E) (1)|, H ≡ C0 + f (P29 , OP2 (6) − 8p2 − p) (P26 , OP2 (3) − 6p) (F1 , C0 + 2f ) (P2 , O

Proof. Recalling that S is a minimal surface of general type it is d2 + 2. pg (S) ≤ 2 Using (11.1) we get d2 (11.9) pg (Y ) ≤ − h0 + 2 2 Riemann Roch theorem for KY + L and (11.1) give L(KY + L) = 2(pg (S) − 2pg (Y ) + q − 1), d2 (11.10) KY (KY + L) = − 2(pg (S) − 2pg (Y ) + q − 1). 2 The proof will be split into two cases, according to the Kodaira dimension of Y. Case 1. κ(Y ) ≥ 0. Ampleness of L and H and (11.10) give d2 (11.11) 2 + 2pg (Y ) − pg (S) ≤ q ≤ + 1 + 2pg (Y ) − pg (S). 4 Combining (11.11) with (11.9) it follows that the following cases may occur: Case d2 pg (Y ) pg (S) i) 4 1 4 ii) 6 2 5 iii) 6 1 4

q h0 0 3 1 3 0 3

In case i) it is KY H = 0 and thus KY must be numerically trivial, L2 = 2 and g(Y, L) = 2. From [23] (15.2) it follows that Y is a K3 surface as in case 2 of the statement. In case ii), since KY L = 0 would imply KY numerically trivial and thus KY2 = 0, (11.10) gives L2 = 1, LKY = 1, KY2 = 0, g(Y, L) = 2. Using [23] (15.2) and (15.9) one can see that there are no such surfaces.

20


In case iii) an argument exactly as the one used above in case ii) shows that there are no such surfaces. Case 2. κ(Y ) = −∞. It is pg (Y ) = 0 and thus from (11.9) it follows that the following cases may occur: Case d2 pg (Y ) pg (S) h0 i) 2 0 3 3 ii) 4 0 3 3 iii) 4 0 4 4 iv) 6 0 3 3 v) 6 0 4 4 vi) 6 0 5 5 It is also χ(OS ) ≥ 1 so q ≤ pg (S). Let us assume first that L2 ≤ 4. One can check, analogously to what it is done in the proof of Proposition 11.4, that Y cannot be P2 or a P1 -bundle. Therefore (KY + H)2 = α ≥ 0. Let β = 4 − L2 ≥ 0. From (11.10) it follows that d2 1 = 2pg (S) + 2q − 3 + (α + β). 2 4 Because pg (S) ≥ 3, the above equality gives case iv) with q = 0 as the only possibility. Therefore g(Y, H) = 2 and the usual [23] (15.2) gives case 5 in the statement. Let us now assume L2 ≥ 5. In this case H is ample and spanned by Theorem 11.1. In case i) it follows that g(Y, H) = 0 and thus we have case 1 in the statement. In case ii) it is g(Y, H) = 1 − q and H 2 = 2. If q = 1 then g(Y, H) = 0 and therefore (Y, H) = (Q, OQ (1)) which is a contradiction because q(Q) = 0. Therefore it must be q = 0 and thus g(Y, H) = 1. Hence (Y, H) is a Del Pezzo surface as in case 3 of the statement. In case iii) it is easy to see that it must be q = g(Y, H) = 0 and H 2 = 2, so that (Y, H) is a quadric in P3 , case 4 in the statement. In case iv) it is g(Y, H) = 2 − q. Since genus zero varieties are all rational, it cannot be q = 2. If q = 1 then g(Y, H) = 1 and thus we must be in case 6 of the statement (see e.g. [23, Theorem 12.3]). If q = 0 then g(Y, H) = 2 and from [23] (15.2) and (15.10) we get case 7 in the statement. In case v) it is g(Y, H) = 1 − q. Since genus zero manifolds are rational it must be q = 0 and g(Y, H) = 1 and thus (Y, H) is a Del Pezzo surface as in case 8 in the statement. In case vi) it is g(Y, H) = −q = 0 and thus it is immediate to see that (Y, H) is as in case 9 in the statement. ¤


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