A bound on the Castelnuovo-Mumford regularity for curves

Math. Ann. 322, 69–74 (2002)

Mathematische Annalen

Digital Object Identifier (DOI) 10.1007/s002080100265

A bound on the Castelnuovo-Mumford regularity for curves Atsushi Noma Received: 29 May 2000 / Published online: 24 September 2001 – © Springer-Verlag 2001 Abstract. Recall that a projective curve in Pr with ideal sheaf I is said to be n-regular if H i (Pr , I ⊗ OPr (n − i)) = 0 for every integer i > 0 and that in this case, it is cut out schemetheoretically by equations of degree at most n. The purpose here is to show that an irreducible, reduced, projective curve of degree d and large arithmetic genus pa satisfies a smaller regularity bound than the optimal one d −r +2. For example, if pa ≥ r −2 then a curve is (d −2r +4)-regular unless it is embedded by a complete linear system of degree 2pa + 2. Mathematics Subject Classification (2000): 14H45, 14N05

Let C ⊆ Pr (r ≥ 3) be an irreducible, reduced, projective curve of degree d and geometric genus g defined over an algebraically closed field k of arbitrary characteristic. Assume that C is nondegenerate, namely C is not contained in any hyperplane in Pr . A coherent sheaf F on Pr is said to be n-regular if H i (Pr , F ⊗ OP (n − i)) = 0 for any i > 0. We say that C is n-regular if its ideal sheaf IC of C in Pr is n-regular. By the theory of CastelnuovoMumford [10, Lecture 14], if C is n-regular then C is (n + 1)-regular and C is scheme-theoretically cut out by hypersurfaces of degree n. Gruson, Lazarsfeld and Peskine [8] have shown that every curve C is (d − r + 2)-regular and that C is (d − r + 1)-regular if g ≥ 2 and also classified all (d − r + 1)-irregular curves. Based on an idea in the result for the (d − r + 1)-regularity, Ellia [5] and D’Almeida [3] have shown that C is (d − r)-regular if C is nonhyperelliptic with geometric genus g ≥ r + 2 and g ≥ 3 respectively. For r = 3, further developments were given by D’Almeida [2] and Mukai [9]. The purpose here is to give a bound on the regularity for C of large arithmetic genus, based on the idea in [8, Theorem 2.1], [5] and [3]. The main result of this paper is the following. Theorem 1. Let C ⊆ Pr be a nondegenerate, irreducible, reduced, projective curve of degree d and arithmetic genus pa defined over an algebraically closed A. Noma∗ Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-ku, Yokohama 240-8501 Japan (e-mail: [email protected]) ∗ Partially supported by Grant-in-Aid for Encouragement of Young Scientists, Japan Society for

the Promotion of Science.

70

A. Noma

field of arbitrary characteristic. Let l be a positive integer satisfying pa ≥ l and r ≥ l + 2. Then C is (d − r + 2 − l)-regular and d ≥ r + l + 1 unless C is a curve embedded by a complete linear system of degree d ≥ 2pa + 2 and l = pa . Theorem 1 for l = 1 is [8, Theorem 2.1 and Remark p.501], since pa ≥ g. Theorem 1 for l = 2 implies the results of [5] and [3] in case r ≥ 4 without assuming that C is nonhyperelliptic. In particular, for l = r − 2, we have the following. Corollary 2. Let C ⊆ Pr (r ≥ 3) be a nondegenerate, irreducible, reduced, projective curve of degree d and arithmetic genus pa . If pa ≥ r − 2, then C is (d − 2r + 4)-regular and d ≥ 2r − 1 unless C is a curve embedded by a complete linear system of degree d = 2pa + 2. Theorem 1 is a consequence of the proposition below. Proposition 3. Let C ⊆ Pr (r ≥ 3) be a nondegenerate, irreducible, reduced, projective curve of degree d and arithmetic genus pa . Let p : C → Pr be the embedding. Set OC (n) = p ∗ OPr (n) for every integer n and M = p ∗ (ΩP1r ⊗ OPr (1)). Let l be a positive integer with pa ≥ l and h0 (C, OC (1)) ≥ l + 1. Assume that H 0 (C, ∧r−2 M ⊗ ωC ⊗ A∗ ) = 0 for the line bundle A associated with an effective divisor of general smooth (pa −l) points of C and for the dualizing sheaf ωC of C. Then C is (d −r +2−l)regular and d ≥ r + l + 1 unless pa = l and d = r + l. We begin with the proof of Proposition 3 by proving the lemma below, which is generalization of that in [8]. Lemma 4. Let C, OC (n), and l be as in Proposition 3. Let A be the line bundle on C associated with an effective divisor of general smooth (pa − l) points of C. The graded module

H 0 (Pr , p∗ A ⊗ OPr (n)) ( = H 0 (C, A ⊗ OC (n)) ) E := n∈Z

n∈Z

over the homogeneous coordinate ring S := k[T0 , . . . , Tr ] of the projective space Pr admits a minimal graded free resolution with the following numerical type: (1) 0 → S nr−1 (−r) ⊕ S l (−r − 1) → S nr−2 (−r + 1) ⊕ S mr−2 (−r) → · · · → S n1 (−2) ⊕ S m1 (−3) → S ⊕ S d−r−l (−1) ⊕ S m0 (−2) → E → 0. Here S n (e) denotes n copies of the eth twist S(e) of S. (2) d − r − l ≥ 0. (3) m1 = h0 (C, ∧r−2 M ⊗ ωC ⊗ A∗ ).

A bound on the Castelnuovo-Mumford regularity for curves

71

(4) If mi = 0 for some i > 0 then mj = 0 for all 0 ≤ j < i. Proof. First note that for a torsion-free sheaf L of rank 1 with nonzero global sections, if x is a general point of an open subset consisting of smooth points of C, then 0 → H 0 (L ⊗ OC (−x)) → H 0 (L) → H 0 (L ⊗ k(x)) → 0 is exact. Hence if A is the line bundle on C associated with an effective divisor of general smooth (pa − l) points of C, then h0 (ωC ⊗ A∗ ) = pa − (pa − l) = l since ωC is a torsion-free of rank 1 (see [1, (I.2.8)]). By duality and Riemann-Roch Theorem, h0 (A) = h0 (ωC ⊗A∗ )+deg A+1−pa = 1. Moreover h0 (ωC ⊗A∗ ⊗OC (−n)) = h1 (A ⊗ OC (n)) = 0 for all n > 0. Indeed, if not, there is a nonzero map OC (n) → ωC ⊗ A∗ and hence l + 1 ≤ h0 (OC (n)) ≤ h0 (ωC ⊗ A∗ ) = l, contradiction. By the same argument, we have h0 (A ⊗ OC (−n)) = 0 for all n > 0 (see [8, Lemma 2.2]). We will look at the numerical type of a minimal graded free resolution P• of E with Pi = ⊕j S βi j (−i − j ) according to [6] and [7]. For the irrelevant maximal ideal m, by calculating Tor Si (E, S/m) from the resolution of E and from the Koszul complex · · · → ∧i V ⊗ S(−i) → · · · → S → S/m → 0 for V := H 0 (OPr (1)) respectively, we know that the set [Tor Si (E, S/m)]i+j of elements of degree i + j is a k-vector space of dimension βi j and is isomorphic to the homology of ∧i+1 V ⊗ H 0 (A ⊗ OC (j − 1)) → ∧i V ⊗ H 0 (A ⊗ OC (j )) → ∧i−1 V ⊗ H 0 (A ⊗ OC (j + 1)). To see the homology, consider the exact sequences 0 → ∧p M ⊗A⊗OC (q) → ∧p V ⊗A⊗OC (q) → ∧p−1 M ⊗A⊗OC (q +1) → 0 for (p, q) = (i + 1, j − 1), (i, j ), and (i − 1, j + 1) obtained from 0 → M → V ⊗ OC → OC (1) → 0 and its long exact sequence α

∧i+1 V ⊗ H 0 (A ⊗ OC (j − 1)) → H 0 (∧i M ⊗ A ⊗ OC (j )) → H 1 (∧i+1 M ⊗ A ⊗ OC (j − 1)) → ∧i+1 V ⊗ H 1 (A ⊗ OC (j − 1)). Thus the homology group is the cokernel of α and hence βi j = dimk Coker(α). Since h0 (A) = 1 and h0 (A ⊗ OC (−n)) = 0 for n > 0, we have β0 0 = 1 and β0 j = 0 for j < 0. Since the multiplication map V ⊗H 0 (A) → H 0 (A⊗OC (1)) is injective and since h0 (A⊗OC (1)) = d −l +1, we have β1 0 = h0 (M ⊗A) = 0 and β0 1 = d − r − l ≥ 0, and hence βi 0 = 0 for all i > 1. The bijectivity of α for i = r implies βr j = 0 for all j . Since ∧i+1 V ⊗ H 1 (A ⊗ OC (j − 1)) = 0 for all j − 1 > 0 and since H 1 (∧i+1 M ⊗ A ⊗ OC (j − 1))∗ ∼ = H 0 (∧r−i−1 M ⊗ ωC ⊗ A∗ ⊗ OC (2 − j )) ⊆ ∧r−i−1 V ⊗ H 0 (ωC ⊗ A∗ ⊗ OC (2 − j )),

72

A. Noma

we have βi j = h1 (∧i+1 M ⊗ A ⊗ OC (j − 1)) for j ≥ 2 and βi j = 0 for j ≥ 3, and in particular, βr−1 2 = h0 (ωC ⊗ A∗ ) = l > 0. Therefore if we set mi = βi 2 and ni = βi 1 , we have (1), (2) and (3). By duality (see for example [4, Theorem A 4.1 and A 4.2]), Ext iS (E, S(−r − 1)) is isomorphic to G := ⊕n∈Z H 0 (ωC ⊗ A∗ ⊗ OC (n)) if i = r − 1 or to 0 otherwise. Thus P•∗ ⊗ S(−r − 1) gives a minimal graded free resolution of G, and hence (4) holds.   Proof of Proposition 3. We will follow the proof of [8, Theorem 2.1]. Let A be the line bundle on C associated with an effective divisor of general smooth (pa − l) points of C. By Lemma 4 together with our assumption, p∗ A admits a resolution as follows: n

n

m

r−2 r−2 l 0 → OPr−1 (−r) → · · · r (−r) ⊕ OPr (−r − 1) → OPr (−r + 1) ⊕ OPr

v

→

(−1) OPn2r (−3) ⊕ OPmr2 (−4) → OPn1r (−2) → OPr ⊕ OPd−r−l r

→

p∗ A → 0.

(−1) is the If s is a nonzero section of H 0 (p∗ A) and if u : OPn1r (−2) → OPd−r−l r composite of v and the projection to the second factor, we have the following commutative diagram with exact rows and columns: 0 0 ↓ ↓ O Pr → OC →0 ↓ ↓s v (−1) → p →0 0 → K → OPn1r (−2) → OPr ⊕ OPd−r−l r ∗A  ↓ ↓ u (−1) → Coker(s) →0 0 → N → OPn1r (−2) → OPd−r−l r ↓ ↓ 0 0, where K and N are the kernels of v and u, respectively. Hence N/K ∼ = IC . By the resolution of p∗ A, K is 4-regular and hence (d − r + 3 − l)-regular if d − r − l ≥ 1. Moreover if pa > l then Coker(s) = 0 and hence d − r − l ≥ 1. Since N is (d −r +2−l)-regular by Lemma 5 below, IC is (d −r +2−l)-regular unless pa = l and d = r + l.   f

Lemma 5. Let u : OPe r (−2) → OPr (−1) be a morphism on Pr whose cokernel is supported on a set of dimension ≤ 0. Then the kernel of u is (f + 2)-regular.

A bound on the Castelnuovo-Mumford regularity for curves

73

f

Proof. ([8]) Set E = OPe r (−2) and F = OPr (−1). Consider the Eagon-Northcott complex 0 → ∧e E ⊗ Syme−f −1 (F)∗ ⊗ ∧f F ∗ → · · · → ∧f +2 E ⊗ F ∗ ⊗ ∧f F ∗ w

u

→ ∧f +1 E ⊗ ∧f F ∗ → E → F → 0, for u (see [8, (0.5)], [4, §A.2.6.1]), which is exact away from the support of the cokernel of u. On the other hand, [8, Lemma 1.6] implies immediately that a , coherent sheaf G on Pr fitting into a complex P• → G → 0 (Pl = ⊕il OPr (−al il )) is t-regular if it is exact away from a set of dimension ≤ 1, if , is surjective, and if al il − l ≤ t for all l (0 ≤ l ≤ r − 1) and il . Thus the image Im(w) of w is (f + 2)-regular, since ∧f +i+1 E ⊗ Symi (F)∗ ⊗ ∧f F ∗ is a sum of copies of OPr (−(f + i + 2)). Consequently the kernel Ker(u) of u is (f + 2)-regular, since Ker(u)/Im(w) is supported on a set of dimension ≤ 0.   To obtain Theorem 1 from Proposition 3, we need the following lemma, which is a modification of [8, Remark (2) after Lemma 1.7]. Lemma 6. Let C ⊆ Pr and p be as in Proposition 3. Let x0 , . . . , xr−1 ∈ C be r distinct points such that p(x0 ), . . . , p(xr−1 ) span a hyperplane in Pr cutting out d distinct smooth points of C. Then M = p ∗ (ΩP1r ⊗ OPr (1)) admits a filtration by vector bundles M = F r ⊃ F r−1 ⊃ · · · ⊃ F 1 ⊃ F 0 = 0 such that F r−j /F r−j −1 ∼ = OC (−Bj ) (j = 0, . . . , r − 1) for effective divisors Bj on C whose support Supp(Bj ) is on smooth locus of C and contains xj . Proof. After a change of coordinates T0 , . . . , Tr of Pr , we may assume that the linear space Lj spanned by p(x0 ), . . . , p(xj ) in Pr is defined by Tr = Tr−1 = · · · = Tj +1 = 0. Set Dj = p −1 (Lj ). The image of the evaluation map ⊕ri=j +1 OC · Ti → OC (1) is OC (1) ⊗ OC (−Dj ). Let F r−j −1 be the kernel of the evaluation map and set F r = M. If we set D−1 = 0 and Bj = Dj −Dj −1 for j = 0, . . . , r−1, then xj ∈ Supp(Bj ) and F r−j /F r−j −1 ∼   = OC (−Bj ), as required. Proof of Theorem 1. Since C is nondegenerate and r ≥ l+2, we have h0 (OC (1)) ≥ l + 1. By Reg(C) we denote the smooth locus of C. By the remark in the proof of Lemma 4, the open subset W  := {(y1 , . . . , yr−2 ) ∈ Reg(C)r−2 |h0 (OC (−y1 − · · · − yr−2 ) ⊗ ωC ⊗ A∗ ) = 0} of Reg(C)r−2 is nonempty. For the projection pi1 ,...,ir−2 : Reg(C)r → Reg(C)r−2 to (i1 , . . . , ir−2 )-th factors, W := ∩i1 ,...,ir−2 pi−1 (W  ) is an open dense sub1 ,...,ir−2 r set of Reg(C) . By Bertini’s theorem, there exists an open dense subset U of

74

A. Noma

Reg(C)r consisting of points (y1 , . . . , yr ) such that p(y1 ), . . . , p(yr ) span a hyperplane cutting out d distinct points of Reg(C). For (x0 , . . . , xr−1 ) ∈ U ∩ W , we consider the filtration in Lemma 6. If 0 ≤ i1 < i2 < · · · < ir−2 ≤ r − 1, then H 0 (C, OC (−xi1 − · · · − xir−2 ) ⊗ ωC ⊗ A∗ ) = 0 by construction, and hence H 0 (C, OC (−Bi1 − · · · − Bir−2 ) ⊗ ωC ⊗ A∗ ) = 0. Thus we have the required vanishing, and therefore C is (d −r +2−l)-regular unless pa = l and d = r +pa by Proposition 3. If pa = l, d = r + pa , and r ≥ l + 2, then d = r + pa ≥ 2pa + 2, and hence h1 (OC (1)) = h0 (ωC ⊗ OC (−1)) = 0 and h0 (OC (1)) = d + 1 − pa = r + 1. Therefore p : C → Pr is the embedding by the complete linear system associated with OC (1).   Remark. If a curve C of genus pa > 0 is embedded by a complete linear system of degree d ≥ 2pa + 2 in Pr , then C ⊆ Pr is 3-regular but not 2-regular. Indeed, if we set l = pa , then we have d − r − l = 0 and r ≥ l + 2. By Lemma 6, we have H 0 (C, ∧r−2 M ⊗ ωC ) = 0 for C ⊆ Pr . Therefore by resolving A := OC with Lemma 4, we know that OC is 2-regular but not 1-regular and that C is projectively normal. Hence IC is 3-regular but not 2-regular. Acknowledgement. I should like to thank Professor Sijong Kwak for stimulating discussion about this matter.

References 1. A. Altman, S. Kleiman: Introduction to Grothendieck duality theory. Lecture Notes in Math. 146. Springer, Berlin New York 1970 2. J. D’Almeida: Courbes de l’espace projectif: S´eries lin´eaires incompl`etes et multis´ecantes. J. Reine Angew. Math. 370, 30–51 (1986) 3. J. D’Almeida: R´egularit´e des courbes de Pr . C. R. Acad. Sci. Paris S´er. I Math. 309, 357–358 (1989) 4. D. Eisenbud: Commutative algebra with a view toward algebraic geometry. Graduate Texts in Math. 150. Springer, Berlin New York 1994 5. Ph. Ellia: Une remarque sur un th´eor`eme de Gruson-Lazarsfeld-Peskine. Arch. Math. (Basel) 48, 406–408 (1987) 6. M. Green: Koszul cohomology and the geometry of projective varieties. J. Differential Geom. 19, 125–171 (1984) 7. M. Green, R. Lazarsfeld: Some results on the syzygies of finite sets and algebraic curves. Compositio Math. 67, 301–314 (1988) 8. L. Gruson, R. Lazarsfeld, C. Peskine: On a theorem of Castelnuovo, and the equations defining space curves. Invent. Math. 72, 491–506 (1983) 9. S. Mukai: Equations defining a space curve. Preprint 1998 10. D. Mumford: Lectures on curves on an algebraic surface. Ann. of Math. Studies 59 (1966)