COMPLETE INTERSECTIONS IN PROJECTIVE SPACE AND SMALL WEIGHT CODEWORDS OF DUALS OF ALGEBRO-GEOMETRIC CODES

COMPLETE INTERSECTIONS IN PROJECTIVE SPACE AND SMALL WEIGHT CODEWORDS OF DUALS OF ALGEBRO-GEOMETRIC CODES

EDOARDO BALLICO

We give a uniformity property of finite complete intersection sets: if S ⊂ Pⁿ is the complete intersection of n hypersurfaces of degree m1, . . . , mn, then h¹(Pⁿ,IS⁰(m1+· · ·+mn−n−1)) = 0 for allS⁰(S.

AMS 2010 Subject Classification: 14N05.

Key words: affine code, evaluation code, complete intersection, dual code, alge- braic-geometric code.

1. INTRODUCTION

Fix a primepand ap-powerq. We recall that an affine [n, k]-codeC over Fq is usually given in the following way. Fix positive integers m, n, distinct pointsP₁, . . . , P_nof the affine spaceF^mq and ak-dimensional linear subspaceV ofFq[x₁, . . . , x_m] such that nof ∈V\{0}vanishes at all pointsP₁, . . . , P_n. Fix a basisf1, . . . , f_kofV. Thek×nmatrix (fi(Pj)) gives an injective linear map F^k → Fⁿq, i.e., this matrix is the generator matrix of an [n, k]-code C. Often these codes are called evaluation codes. The dual code C^∨ is the [n,(n−k)]- code whose words are the elements of Fⁿq orthogonal to the words of C with respect to the canonical inner product, i.e., (a1, . . . , an)∈Fⁿ_q is a word ofC^∨ if and only ifa1b1+· · ·+a_nbn= 0 for all words (b1, . . . , bn) ofC(i.e., the generator matrix of C^∨ is the parity check matrix of C, and conversely). A. Couvreur proved that quite often it is easier to compute the minimum distance of C^∨ and classify all the codewords ofC^∨with small weights (not only the ones with minimum distance) than to solve the corresponding problems for C ([2]; see also [3], which uses [2] for codes arising from the Hermitian curve). As clear from [2] in evaluation codes arising fromH⁰(Pⁿ,O_Pn(m)) very simple objects (lines, small degree plane curves, finite sets which are complete intersections) are often useful. Our aim is to show that the minimal free resolutions of these objects is a key tool even to study their subsets. As in [2] the first step is to solve the problem over the algebraic closure of the field we are interested here.

REV. ROUMAINE MATH. PURES APPL.,56(2011),3, 181–184

182 Edoardo Ballico 2

Hence from now on we work over an algebraically closed field. The property obtained for the setS introduced in Theorem 1.1 and Proposition 1.2 below is a uniformity property for its subsets. Indeed, in this case the non-trivial part of it says that all proper subsets of S have good cohomology with respect to the line bundle O_Pn(m₁ +· · ·+m_n−n−1); in this statement the essential part is the case of all S⁰⊂S such that](S⁰) =](S)−1.

For any integraln-dimensional projective varietyM and anynline bun- dlesL1, . . . , LnonM letL1. . . Lndenote the intersection product of these line bundles in the sense of [4], Chapter 2 and Chapter 6, or of [7], Remark 1.1.13.

For instance, if M = Pⁿ and L_i = O_Pⁿ(m_i), then L₁. . . L_n = Qn

i=1m_i. We prove the following result.

Theorem 1.1. Let M be an integral and locally Cohen-Macaulay n- dimensional projective variety,n≥2. Fixnample line bundlesL1, . . . , Ln(we allow the case L_i ∼=L_j for some i6=j). For any finite subset E ⊆ {1, . . . , n}

set L^⊗E := ⊗i∈ELi. Assume the existence of Di ∈ |L_i|, 1 ≤i≤ n such that the set S:= (D1∩ · · · ∩Dn)_red is finite and has cardinality L1. . . Ln. Assume Hⁱ(M, ωM⊗L^E) = 0 for all i∈ {1, . . . , n−1} and anyE ⊆ {1, . . . , n}. Then h¹(M,I_S⊗ω_M⊗(⊗ⁿ_i=1L_i)) = 1andh¹(M,I_S⁰⊗ω_M⊗(⊗ⁿ_i=1L_i)) = 0for every S⁰ (S.

Taking M ∼= Pⁿ in the statement of Theorem 1.1 we get the following result.

Proposition 1.2. Fix integers n≥2 and mi >0, 1≤i≤n. Let Di⊂ Pⁿ, 1≤i≤n, be degree m_i hypersurfaces such thatS := (D₁∩. . .∩D_n)_red is a finite set with cardinality Qn

i=1mi. Then h¹(I_S(m1+· · ·+mn−n−1)) = 1 and h¹(I_S⁰(m1+· · ·+mn−3)) = 0 for everyS⁰ (S.

Proof of Theorem1.1. Since ](S) =L1. . . Ln and the intersection of the effective divisors D₁∩. . .∩D_nis finite, S is the scheme-theoretic intersection of the divisors D1, . . . , Dn ([4], Example 2.4.8). Hence the Koszul complex (1) 0→L^{1,...,n}∨→ · · · ⊕E⊂{1,...,n}:](E)=iL^E∨· · · → ⊕ⁿ_i=1L_i^∨ → I_S→0 is exact. We tensor it with ω_M ⊗(⊗ⁿ_i=1L_i) and break it into short exact se- quences. For any R∈Pic(M) we have hⁿ(M, ωM⊗R) =h⁰(M, R^∨) for every line bundleRonM (the part of [1], caser :=n,p:=r,F :=Rof the theorem at page 1, which does not require that M is locally Cohen-Macaulay). Taking R = O_M we get hⁿ(M, ω_X) = h⁰(M,O_M) = 1. Taking R = ⊗ⁿ_i=1L_i we get hⁿ(M, ωM⊗(⊗ⁿ_i=1Li)) =h⁰(M,O_M(−D₁− · · · −Dn)). Since the line bundles L_i, 1 ≤ i ≤ n, are ample, we have L₁. . . L_n > 0 ([7], Example 1.2.5, or [5], p. 30). Hence S 6=∅. Therefore, D1+· · ·+Dn6=∅. Since M is integral and

3 Complete intersections in projective space 183

projective, we get h⁰(M,O_M(−D₁ − · · · −Dn)) = 0. Hence from the coho- mology exact sequences of all the short exact sequences we just obtained we get h¹(M,I_S⊗ωM ⊗L1⊗ · · ·Ln) = 1.

For any E ⊆ S, let ρ_E : H⁰(M, ω_M ⊗L1 ⊗. . . Ln) → H⁰(E, ω_M ⊗ L₁ ⊗ · · ·L_n|E) denote the restriction map. Since we assumed h¹(M, ω_M ⊗ (⊗ⁿ_i=1Li)) = 0, ρE is surjective if and only ifh¹(M,I_E⊗ωM⊗(⊗ⁿ_i=1Li)) = 0.

Since h¹(M,I_S⊗ωM⊗(⊗ⁿ_i=1Li)) = 1, the linear mapρS has corank 1. Hence ρ_E is surjective for every E ⊂S such that ](E) ≤ ](S)−2. Therefore only the case ](S⁰) = ](S)−1 is non-trivial. The set |I_S ⊗L₁| · · · |I_S ⊗L_n| is non-empty, because it contains (D1, . . . , Dn). Fix a general (A1, . . . , An) ∈

|I_S⊗L₁| · · · |I_S⊗L_n|and setY :=A₂∩. . .∩A_n(scheme-theoretic intersection).

Since the scheme Y is the intersection of n−1 effective Cartier divisors ofY, each irreducible component ofY_redhas dimension≥1. Notice thatS ⊆Y∩A₁. Since ](S) =L₁. . . L_n, we get thatSis the scheme-theoretic intersection ofY and A1. SinceL1 is ample, we also get that each irreducible component T of Y_redis one-dimensional and it contains at least one point ofS (use thatL₁|T is ample and hence deg(L1|T)>0). SinceS is the scheme-theoretic intersection of A₁ and Y, thenY is smooth at each point ofS,M is smooth at each point of S, and each one-dimensional component of Y appears with multiplicity 1 in Y. SinceY is the scheme-theoretic intersection ofn−1 Cartier divisors of M, M is locally Cohen-Macaulay and dim(Y) = 1, then Y is locally Cohen- Macaulay. Since Y is generically reduced and locally Cohen-Macaulay, then Y is reduced. Since each L_i, i ≥ 2, is ample, Y is connected ([6], Proofs of Corollaries III.7.8 and III.7.9 and our assumption that M is locally Cohen- Macaulay). The adjunction formula gives ω_S = ω_Y ⊗(L₁|Y). Fix S⁰ ⊂ S such that ](S⁰) = ](S)−1. Set {P} := S \S⁰. Since Y is smooth at P, the divisor P is a Cartier divisor of Y. Since Y is reduced and connected, we have h⁰(Y,O_Y(−P)) = 0, i.e., h¹(Y, ω_Y(P)) = 0 ([1], Theorem 1 in the Introduction). Therefore, the exact sequence

(2) 0→ω_Y(P)→ω_Y ⊗(L1|Y)→(ω_Y ⊗(L1|Y))|S⁰ →0

gives the surjectivity of the restriction mapβ :H⁰(Y, ω_Y ⊗(L₁|Y))→H⁰(S⁰, (ω_Y ⊗ (L₁|Y))|S⁰). Therefore, ρ_S⁰ is surjective if the restriction map α : H⁰(M, ωM⊗(⊗ⁿ_i=1Li))→H⁰(Y, ωY ⊗(L1|Y)) is surjective. The equations of A₂, . . . , A_n give the exact sequence

(3) 0→L^{2,...,n}∨→ · · · ⊕E⊂{2,...,n},](E)=iL^E∨· · · → ⊕ⁿ_i=2L_i^∨ → I_Y →0 (Koszul complex). We tensor it with ω_M ⊗(⊗ⁿ_i=1Li) and break it into short exact sequences. Since hⁿ(M, ωM ⊗L1) = h⁰(M, L^∨₁) = 0 (duality and the ampleness ofL₁), we obtainh¹(M,I_Y ⊗ω_M⊗(⊗ⁿ_i=1L_i)) = 0. Therefore,α is surjective.

184 Edoardo Ballico 4

REFERENCES

[1] A. Altman and S. Kleiman, Introduction to Grothendieck’s duality theory. Lect. Notes in Math.146, Springer-Verlag, 1970.

[2] A. Couvreur,The dual minimum distance of arbitrary dimensional algebraic-geometric codes. arXiv:0905.2345v3, J. Algebra (to appear).

[3] C. Fontanari and C. Marcolla,On the geometry of small weight codewords of dual alge- braic geometric codes. arXiv:1104.1320v1.

[4] W. Fulton,Intersection Theory. Springer-Verlag, Berlin, 1984.

[5] R. Hartshorne, Ample Subvarieties of Algebraic Geometry. Lect. Notes in Math. 156, Springer-Verlag, Berlin, 1970.

[6] R. Harshorne,Algebraic Geometry. Springer-Verlag, Berlin, 1977.

[7] R. Lazarsfeld,Positivity in Algebraic Geometry I. Springer, Berlin, 2004.

Received 30 May 2011 University of Trento

Department of Mathematics 38123 Povo (TN), Italy

ballico@science.unitn.it