Nondegenerate Curves of Low Genus over Small Finite Fields

In a previous paper, we proved that over a finite field k of sufficiently large cardinality, all curves of genus at most 3 over k can be modeled by a bivariate Laurent polynomial that is nondegenerate with respect to its Newton polytope. In this paper, we prove that there are exactly two curves of genus at most 3 over a finite field that are not nondegenerate, one over F2 and one over F3. Both of these curves have remarkable extremal properties concerning the number of rational points over various extension fields. Let k be a perfect field with algebraic closure k. To a Laurent polynomial f = ∑ (i,j)∈Z cijx y ∈ k[x, y], we associate its Newton polytope ∆(f), the convex hull in R of the points (i, j) ∈ Z for which cij 6= 0. An irreducible Laurent polynomial f is called nondegenerate with respect to its Newton polytope if for all faces τ ⊂ ∆(f) (vertices, edges, and ∆(f) itself), the system of equations (∗) f |τ = x ∂f |τ ∂x = y ∂f |τ ∂y = 0 has no solution in k ∗2 , where f |τ = ∑ (i,j)∈Z∩τ cijx y. A curve C over k is called nondegenerate if it is birationally equivalent over k to a curve defined by a Laurent polynomial f ∈ k[x, y] that is nondegenerate with respect to its Newton polytope. For such a curve, a vast amount of geometric information is encoded in the combinatorics of ∆(f). For example, the (geometric) genus of C is equal to the number lattice points (points in Z) lying in the interior of ∆(f). Owing to this connection, nondegenerate curves have become popular objects of study in explicit algebraic geometry. (See e.g. Batyrev [1] and the introduction in our preceding work [5] for further background and discussion.) In a previous paper [5], we gave a partial answer to the natural question: Which curves are nondegenerate? Theorem. Let C be a curve of genus g over k. Suppose that one of these conditions holds: (i) g = 0; (ii) g = 1 and C(k) 6= ∅; (iii) g = 2, 3, and either 17 ≤ #k < ∞, or #k = ∞ and C(k) 6= ∅; (iv) g = 4 and k = k. Then C is nondegenerate. If g ≥ 5, then the locus Mnd g of nondegenerate curves inside the coarse moduli space of curves of genus g satisfies dimMnd g = 2g + 1, except for g = 7 where Date: July 14, 2009.