Intrinsicness of the Newton polygon for smooth curves on ℙ¹× ℙ¹

2013/04/18 by Castryck, Wouter, Cools, Filip
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1304.4997

Abstract

Let C be a smooth projective curve in ℙ¹× ℙ¹ of genus g≠ 4, and assume that it is birationally equivalent to a curve defined by a Laurent polynomial that is non-degenerate with respect to its Newton polygon Δ. Then we show that the convex hull Δ⁽¹⁾ of the interior lattice points of Δ is a standard rectangle, up to a unimodular transformation. Our main auxiliary result, which we believe to be interesting in its own right, is that the first scrollar Betti numbers of Δ-non-degenerate curves are encoded in the combinatorics of Δ⁽¹⁾, if Δ satisfies some mild conditions.

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