On nondegeneracy of curves

Castryck, Wouter; Voight, John

doi:10.2140/ant.2009.3.255

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On nondegeneracy of curves

Wouter Castryck and John Voight

Vol. 3 (2009), No. 3, 255–281

DOI: 10.2140/ant.2009.3.255

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 $g \leq 4$ 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 $g \geq 2$ . Then we show that $dim ℳ_{g}^{nd} = min (2 g + 1, 3 g - 3)$ , except for $g = 7$ where $dim ℳ_{7}^{nd} = 16$ ; thus, a generic curve of genus $g$ is nondegenerate if and only if $g \leq 4$ .

Keywords

nondegenerate curve, toric surface, Newton polytope, moduli space

Mathematical Subject Classification 2000

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/

Vol. 3, No. 3, 2009