A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.Comment: Published in the proceedings of 2017 Southern Regional Algebra Conferenc

Intrinsicness of the Newton polygon for smooth curves on P1×P1{\mathbb {P}}^1\times {\mathbb {P}}^1 P 1 × P 1

Let C be a smooth projective curve in of genus , 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 . Then we show that the convex hull of the interior lattice points of 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 -non-degenerate curves are encoded in the combinatorics of , if satisfies some mild conditions