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
Milestones
Received: 11 April 2008
Revised: 22 December 2008
Accepted: 17 February 2009
Published: 1 May 2009
Authors
 Wouter Castryck Katholieke Universiteit Leuven Departement Elektrotechniek (ESAT) Afdeling SCD – COSIC Kasteelpark Arenberg 10 B-3001 Leuven (Heverlee) Belgium John Voight Department of Mathematics and Statistics University of Vermont 16 Colchester Ave Burlington, VT 05401 United States http://www.cems.uvm.edu/~voight/