Discriminants of Toric Varieties

TL;DR

This paper investigates the properties of the discriminant subvariety of singular sections in linear systems on smooth toric varieties, providing bounds, classifications, and degree calculations for specific families.

Contribution

It offers new bounds and classifications for the dimension of discriminants and analyzes their degrees in polarized toric varieties, advancing understanding of their geometric properties.

Findings

Classification of pairs with low-dimensional discriminants

Bounds on discriminant dimensions and their differences

Minimal degree values for discriminants in certain families

Abstract

We study the subvariety of singular sections, the discriminant, of a base point free linear system $|L|$ on a smooth toric variety $X$. On one hand we describe pairs $(X,L)$ for which the discriminant is of low dimension. Precisely, we collect some bounds on this dimension and classify those pairs whose dimension differs the bound less than or equal to two. On the other hand we study the degree of the discriminant for some relevant families on polarized toric varieties, describing their minimal values and the region of the ample cone where this minimal is attained.