aa r X i v : . [ m a t h . AG ] F e b A LOWER BOUND FOR χ ( O S ) VINCENZO DI GENNARO
Abstract.
Let ( S, L ) be a smooth, irreducible, projective, complex surface,polarized by a very ample line bundle L of degree d >
25. In this paperwe prove that χ ( O S ) ≥ − d ( d − χ ( O S ) = − d ( d −
6) if and only if d is even, the linear system | H ( S, L ) | embeds S ina smooth rational normal scroll T ⊂ P of dimension 3, and here, as a divisor, S is linearly equivalent to d Q , where Q is a quadric on T . Moreover, this isequivalent to the fact that the general hyperplane section H ∈ | H ( S, L ) | of S is the projection of a curve C contained in the Veronese surface V ⊆ P , froma point x ∈ V \ C . Keywords : Projective surface, Castelnuovo-Halphen’s Theory, Rational normalscroll, Veronese surface.
MSC2010 : Primary 14J99; Secondary 14M20, 14N15, 51N35. Introduction
In [6], one proves a sharp lower bound for the self-intersection K S of the canonicalbundle of a smooth, projective, complex surface S , polarized by a very ample linebundle L , in terms of its degree d = deg L , assuming d >
35. Refining the lineof the proof in [6], in the present paper we deduce a similar result for the Eulercharacteristic χ ( O S ) of S [1, p. 2], in the range d >
25. More precisely, we provethe following:
Theorem 1.1.
Let ( S, L ) be a smooth, irreducible, projective, complex surface,polarized by a very ample line bundle L of degree d > . Then: χ ( O S ) ≥ − d ( d − . The bound is sharp, and the following properties are equivalent.(i) χ ( O S ) = − d ( d − ;(ii) h ( S, L ) = 6 , and the linear system | H ( S, L ) | embeds S in P as a scrollwith sectional genus g = d ( d −
6) + 1 ;(iii) h ( S, L ) = 6 , d is even, and the linear system | H ( S, L ) | embeds S ina smooth rational normal scroll T ⊂ P of dimension , and here S is linearlyequivalent to d ( H T − W T ) , where H T is the hyperplane class of T , and W T theruling (i.e. S is linearly equivalent to an integer multiple of a smooth quadric Q ⊂ T ). By Enriques’ classification, one knows that if S is unruled or rational, then χ ( O S ) ≥
0. Hence, Theorem 1.1 essentially concerns irrational ruled surfaces.In the range d >
35, the family of extremal surfaces for χ ( O S ) is exactly thesame for K S . We point out there is a relationship between this family and theVeronese surface. In fact one has the following: Corollary 1.2.
Let S ⊆ P r be a nondegenerate, smooth, irreducible, projective,complex surface, of degree d > . Let L ⊆ P r be a general hyperplane. Then χ ( O S ) = − d ( d − if and only if r = 5 , and there is a curve C in the Veronesesurface V ⊆ P and a point x ∈ V \ C such that the general hyperplane section S ∩ L of S is the projection p x ( C ) ⊆ L of C in L ∼ = P , from the point x . In particular, S ∩ L is not linearly normal, instead S is.2. Proof of Theorem 1.1
Remark . ( i ) We say that S ⊂ P r is a scroll if S is a P -bundle over a smoothcurve, and the restriction of O S (1) to a fibre is O P (1). In particular, S is ageometrically ruled surface, and therefore χ ( O S ) = K S [1, Proposition III.21].( ii ) By Enriques’ classification [1, Theorem X.4 and Proposition III.21], oneknows that if S is unruled or rational, then χ ( O S ) ≥
0, and if S is ruled withirregularity >
0, then χ ( O S ) ≥ K S . Therefore, taking into account previousremark, when d >
35, Theorem 1.1 follows from [6, Theorem 1.1]. In order toexamine the range 25 < d ≤
35, we are going to refine the line of the argument inthe proof of [6, Theorem 1.1].( iii ) When d = 2 δ is even, then d ( d −
6) + 1 is the genus of a plane curve ofdegree δ , and the genus of a curve of degree d lying on the Veronese surface.Put r + 1 := h ( S, L ). Therefore, | H ( S, L ) | embeds S in P r . Let H ⊆ P r − bethe general hyperplane section of S , so that L ∼ = O S ( H ). We denote by g the genusof H . If 2 ≤ r ≤
3, then χ ( O S ) ≥
1. Therefore, we may assume r ≥ The case r = 4.We first examine the case r = 4. In this case we only have to prove that, for d >
25, one has χ ( O S ) > − d ( d − S is an irrational ruledsurface, so K S ≤ χ ( O S ) (compare with previous Remark 2.1, ( ii )). We argue bycontradiction, and assume also that(1) χ ( O S ) ≤ − d ( d − . We are going to prove that this assumption implies d ≤
25, in contrast with ourhypothesis d > d ( d − − g −
1) + 12 χ ( O S ) = 2 K S , and K S ≤ χ ( O S ), we get: d ( d − − g − ≤ χ ( O S ) . LOWER BOUND FOR χ ( O S ) 3 And from χ ( O S ) ≤ − d ( d −
6) we obtain(2) 10 g ≥ d − d + 10 . Now we distinguish two cases, according that S is not contained in a hypersurfaceof degree < S is not contained in a hypersurface of P of degree < d >
16, by Roth’s Theorem ([12, p. 152], [8, p. 2, (C)]), H is not containedin a surface of P of degree <
5. Using Halphen’s bound [9], we deduce that g ≤ d
10 + d −
25 ( ǫ + 1)(4 − ǫ ) , where d − m + ǫ , 0 ≤ ǫ <
5. It follows that32 d − d + 10 ≤ g ≤ d + 5 d + 10 (cid:18) −
25 ( ǫ + 1)(4 − ǫ ) (cid:19) . This implies that d ≤
25, in contrast with our hypothesis d > S is contained in an irreducible and reducedhypersurface of degree s ≤
4. When s ∈ { , } , one knows that, for d > S isof general type [2, p. 213]. Therefore, we only have to examine the case s = 4. Inthis case H is contained in a surface of P of degree 4. Since d >
12, by Bezout’sTheorem, H is not contained in a surface of P of degree <
4. Using Halphen’sbound [9], and [8, Lemme 1], we get: d − d ≤ g ≤ d . Hence, there exists a rational number 0 ≤ x ≤ g = d d (cid:18) x − (cid:19) + 1 . If 0 ≤ x ≤ , then g ≤ d − d + 1, and from (2) we get320 d − d + 1 ≤ g ≤ d − d + 1 . It follows d ≤
24, in contrast with our hypothesis d > < x ≤
9. Hence, (cid:18) d (cid:19) − g = − d (cid:18) x − (cid:19) < d. By [5, proof of Proposition 2, and formula (2.2)], we have χ ( O S ) ≥ d − d − d − − ( d − (cid:20)(cid:18) d (cid:19) − g (cid:21) > d − d − d − − ( d −
3) 316 d = d − d − d − . Combining with (1), we get d − d − d − d ( d − < , i.e. d − d − d − < . VINCENZO DI GENNARO
It follows d ≤
23, in contrast with our hypothesis d > r = 4. The case r ≥ r ≥
5, by [6, Remark 2.1], we know that, for d >
5, one has K S > − d ( d − r = 5, and the surface S is a scroll, K S = 8 χ ( O S ) = 8(1 − g ), and(3) g = 18 d − d + (5 − ǫ )( ǫ + 1)8 , with d − m + ǫ , 0 < ǫ ≤
3. In this case, by [6, pp. 73-76], we know that,for d > S is contained in a smooth rational normal scroll of P of dimension3. Taking into account that we may assume K S ≤ χ ( O S ) (compare with Remark2.1, ( i ) and ( ii )), at this point Theorem 1.1 follows from [6, Proposition 2.2], when d > ≤ d ≤
30, we refine the analysisappearing in [6]. In fact, we are going to prove that, assuming r = 5, S is a scroll,and (3), it follows that S is contained in a smooth rational normal scroll of P ofdimension 3 also when 26 ≤ d ≤
30. Then we may conclude as before, because [6,Proposition 2.2] holds true for d ≥ S is contained in a threefold T ⊂ P of dimension 3 andminimal degree 3, then T is necessarily a smooth rational normal scroll [6, p. 76].Moreover, observe that we may apply the same argument as in [6, p. 75-76] in orderto exclude the case S is contained in a threefold of degree 4. In fact the argumentworks for d >
24 [6, p. 76, first line after formula (13)].In conclusion, assuming r = 5, S is a scroll, and (3), it remains to exclude that S is not contained in a threefold of degree <
5, when 26 ≤ d ≤ S is not contained in a threefold of degree <
5. Denote by Γ ⊂ P thegeneral hyperplane section of H . Recall that 26 ≤ d ≤ • Case I: h ( P , I Γ (2)) ≥ d >
4, by monodromy [4, Proposition 2.1], Γ shouldbe contained in a reduced and irreducible space curve of degree ≤
4, and so, for d > S should be contained in a threefold of degree ≤ • Case II: h ( P , I Γ (2)) = 1 and h ( P , I Γ (3)) > d >
6, by monodromy, Γ is contained in a reduced and irreduciblespace curve X of degree deg( X ) ≤
6. Again as before, if deg( X ) ≤
4, then S iscontained in a threefold of degree ≤
4. So we may assume 5 ≤ deg( X ) ≤ d ≥
26, by Bezout’s Theorem we have h Γ ( i ) = h X ( i ) for all i ≤
4. Let X ′ be the general plane section of X . Since h X ( i ) ≥ P ij =0 h X ′ ( j ), we have h X (3) ≥ h X (4) ≥
19 [7, pp. 81-87]. Therefore, when d ≥
26, taking into account [7,Corollary (3.5)], we get: h Γ (1) = 4 , h Γ (2) = 9 , h Γ (3) ≥ , h Γ (4) ≥ ,h Γ (5) ≥ , h Γ (6) ≥ min { d, } , h Γ (7) = d. LOWER BOUND FOR χ ( O S ) 5 It follows that: p a ( C ) ≤ + ∞ X i =1 d − h Γ ( i ) ≤ ( d − d − d − d − d − d − , which is < d ( d −
6) + 1 for d ≥
26. This is in contrast with (3). • Case III: h ( P , I Γ (2)) = 1 and h ( P , I Γ (3)) = 4.We have: h Γ (1) = 4 , h Γ (2) = 9 , h Γ (3) = 16 , h Γ (4) ≥ , h Γ (5) ≥ , h Γ (6) = d. It follows that: p a ( C ) ≤ + ∞ X i =1 d − h Γ ( i ) ≤ ( d −
4) + ( d −
9) + ( d −
16) + ( d −
19) + ( d −
24) = 5 d − , which is < d ( d −
6) + 1 for d ≥
26. This is in contrast with (3). • Case IV: h ( P , I Γ (2)) = 0.We have: h Γ (1) = 4 , h Γ (2) = 10 , h Γ (3) ≥ , h Γ (4) ≥ ,h Γ (5) ≥ , h Γ (6) ≥ min { d, } , h Γ (7) = d. It follows that: p a ( C ) ≤ + ∞ X i =1 d − h Γ ( i ) ≤ ( d − d − d − d − d − d − , which is < d ( d −
6) + 1 for d ≥
26. This is in contrast with (3).This concludes the proof of Theorem 1.1.
Remark . ( i ) Let Q ⊆ P be a smooth quadric, and H ∈ |O Q (1 , d − | be asmooth rational curve of degree d [11, p. 231, Exercise 5.6]. Let S ⊆ P be theprojective cone over H . A computation, which we omit, proves that χ ( O S ) = 1 − (cid:18) d − (cid:19) . Therefore, if S is singular, it may happen that χ ( O S ) < − d ( d − − (cid:0) d − (cid:1) is a lower bound for χ ( O S ) for every integral surface.( ii ) Let ( S, L ) be a smooth surface, polarized by a very ample line bundle L ofdegree d . By Harris’ bound for the geometric genus p g ( S ) of S [10], we see that p g ( S ) ≤ (cid:0) d − (cid:1) . Taking into account that for a smooth surface one has χ ( O S ) = h ( S, O S ) − h ( S, O S ) + h ( S, O S ) ≤ h ( S, O S ) = 1 + p g ( S ), from Theorem 1.1we deduce (the first inequality only when d > − (cid:18) d − (cid:19) ≤ χ ( O S ) ≤ (cid:18) d − (cid:19) . VINCENZO DI GENNARO Proof of Corollary 1.2 • First, assume that χ ( O S ) = − d ( d − r = 5. Moreover, S is contained in a nonsingularthreefold T ⊆ P of minimal degree 3. Therefore, the general hyperplane section H = S ∩ L of S ( L ∼ = P denotes the general hyperplane of P ) is contained in asmooth surface Σ = T ∩ L of L ∼ = P , of minimal degree 3.This surface Σ is isomorphic to the blowing-up of P at a point, and, for asuitable point x ∈ V \ L , the projection of P \{ x } on L ∼ = P from x restricts to anisomorphism p x : V \{ x } → Σ \ E, where E denotes the exceptional line of Σ [1, p. 58].Since S is linearly equivalent on T to d ( H T − W T ) ( H T denotes the hyperplanesection of T , and W T the ruling), it follows that H is linearly equivalent on Σ to d ( H Σ − W Σ ) (now H Σ denotes the hyperplane section of Σ, and W Σ the ruling ofΣ). Therefore, H does not meet the exceptional line E = H Σ − W Σ . In fact, since H = 3, H Σ · W Σ = 1, and W = 0, one has:( H Σ − W Σ ) · ( H Σ − W Σ ) = H − H Σ · W Σ + 2 W = 0 . This implies that H is contained in Σ \ E , and the assertion of Corollary 1.2follows. • Conversely, assume there exists a curve C on the Veronese surface V ⊆ P ,and a point x ∈ V \ C , such that H is the projection p x ( C ) of C from the point x .In particular, d is an even number, and H is contained in a smooth surfaceΣ ⊆ L ∼ = P of minimal degree, and is disjoint from the exceptional line E ⊆ Σ. By[3, Theorem (0.2)], S is contained in a threefold T ⊆ P of minimal degree. T isnonsingular. In fact, otherwise, H should be a Castelnuovo’s curve in P [6, p. 76].On the other hand, by our assumption, H is isomorphic to a plane curve of degree d . Hence, we should have: g = d − d + 1 = d − d + 1(the first equality because H is Castelnuovo’s, the latter because H is isomorphicto a plane curve of degree d ). This is impossible when d > S is contained in a smooth threefold T of minimal degree in P .Now observe that in Σ there are only two families of curves of degree even d and genus g = d − d + 1. These are the curves linearly equivalent on Σ to d ( H Σ − W Σ ), and the curves equivalent to d +26 H Σ + d − W Σ . But only in thefirst family the curves do not meet E . Hence, H is linearly equivalent on Σ to d ( H Σ − W Σ ). Since the restriction Pic( T ) → Pic(Σ) is bijective, it follows that S is linearly equivalent on T to d ( H T − W T ). By Theorem 1.1, S is a fortiori linearlynormal, and of minimal Euler characteristic χ ( O S ) = − d ( d − LOWER BOUND FOR χ ( O S ) 7 References [1] Beauville, A.:
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