On nondegeneracy of curves

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 $g\le 4$ over an algebraically closed field is nondegenerate in the above sense. More generally, let ${\mathcal{ℳ}}_g^{\mathrm{\text{ nd}}}$ be the locus of nondegenerate curves inside the moduli space of curves of genus $g\ge 2$. Then we show that $dim{\mathcal{ℳ}}_g^{\mathrm{\text{ nd}}}=min\left(2g+1,3g-3\right)$, except for $g=7$ where $dim{\mathcal{ℳ}}_7^{\mathrm{\text{ nd}}}=16$; thus, a generic curve of genus $g$ is nondegenerate if and only if $g\le 4$.