On nondegeneracy of curves

Abstract

We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an algebraically closed field is nondegenerate in the above sense. More generally, let M g be the locus of nondegenerate curves inside the moduli space of curves of genus g ≥ 2. Then we show that dimM g = min(2g +1, 3g − 3), except for g = 7 where dimM 7 = 16; thus, a generic curve of genus g is nondegenerate if and only if g ≤ 4. Subject classification: 14M25, 14H10 Let k be a perfect field with algebraic closure k. Let f ∈ k[x, y] be an irreducible Laurent polynomial, and write f = ∑ (i,j)∈Z2 cijx y . We denote by supp(f) = {(i, j) ∈ Z : cij 6= 0} the support of f , and we associate to f its Newton polytope ∆ = ∆(f), the convex hull of supp(f) in R. We assume throughout that ∆ is 2-dimensional. For a face τ ⊂ ∆, let f |τ = ∑ (i,j)∈τ cijx y . We say that f is nondegenerate if, for every face τ ⊂ ∆ (of any dimension), the system of equations (1) f |τ = x ∂f |τ ∂x = y ∂f |τ ∂y = 0 has no solutions in k ∗2 . From the perspective of toric varieties, the condition of nondegeneracy can be rephrased as follows. The Laurent polynomial f defines a curve U(f) in the torus Tk = Spec k[x , y], and Tk embeds canonically in the projective toric surface X(∆)k associated to ∆ over k. Let V (f) be the Zariski closure of the curve U(f) inside X(∆)k. Then f is nondegenerate if and only if for every face τ ⊂ ∆, we have that V (f)∩Tτ is smooth of codimension 1 in Tτ , where Tτ is the toric component of X(∆)k associated to τ . (See Proposition 1.2 for alternative characterizations.) Nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry [4]: a wealth of geometric information about V (f) is contained in the combinatorics of the Newton polytope ∆(f). The notion was initially employed by Kouchnirenko [22], who studied nondegenerate polynomials in the context of singularity theory. Nondenegerate polynomials emerge naturally in the theory of sparse resultants [14] and admit a linear effective Nullstellensatz [8, Section 2.3]. They make an appearance in the study of real algebraic curves in maximal position [26] and in the problem of enumerating curves through a set of prescribed points [27]. In the case where k is a finite field, they arise in the construction of curves with many points [6, 23], in the p-adic cohomology theory of Adolphson and Sperber [2], and in explicit methods for computing zeta functions of varieties over k [8]. Despite their utility and seeming Date: 26 December 2008.