On nondegeneracy of curves

TL;DR

This paper investigates the nondegeneracy of algebraic curves with respect to Laurent polynomials, showing that most low-genus curves are nondegenerate and characterizing the dimension of the nondegenerate locus in the moduli space.

Contribution

It proves that all curves up to genus 4 are nondegenerate and determines the dimension of the nondegenerate locus for higher genus, highlighting a special case at genus 7.

Findings

All curves up to genus 4 are nondegenerate.

The nondegenerate locus has dimension min(2g+1,3g-3) for g ≠ 7.

At genus 7, the nondegenerate locus is 16-dimensional.

Abstract

A curve is called nondegenerate if it can be modeled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We show that up to genus 4, every curve is nondegenerate. We also prove that the locus of nondegenerate curves inside the moduli space of curves of fixed genus g > 1 is min(2g+1,3g-3)-dimensional, except in case g=7 where it is 16-dimensional.