Abstract

We study the conditions under which an algebraic curve can be modeled 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^nd be the locus of nondegenerate curves inside the moduli space of curves of genus g ≥ 2. Then we show that dimM_g^nd = min(2g + 1,3g − 3), except for g = 7 where dimM₇^nd = 16; thus, a generic curve of genus g is nondegenerate if and only if g ≤ 4.

Keywords

nondegenerate curve, toric surface, Newton polytope, moduli space

Publication

Received: 11 April 2008
Accepted: 17 February 2009

Authors