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 4 over an algebraically closed field is nondegenerate in the above sense. More generally, let g nd be the locus of nondegenerate curves inside the moduli space of curves of genus g 2. Then we show that dimg nd = min(2g + 1,3g 3), except for g = 7 where dim7 nd = 16; thus, a generic curve of genus g is nondegenerate if and only if g 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/