#### Vol. 3, No. 3, 2009

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On nondegeneracy of curves

### Wouter Castryck and John Voight

Vol. 3 (2009), No. 3, 255–281
##### 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 be the locus of nondegenerate curves inside the moduli space of curves of genus $g\ge 2$. Then we show that , except for $g=7$ where ; thus, a generic curve of genus $g$ is nondegenerate if and only if $g\le 4$.

##### Keywords
nondegenerate curve, toric surface, Newton polytope, moduli space
Primary: 14M25
Secondary: 14H10