Discriminants of toric varieties

Abstract

We study the subvariety of singular sections, the discriminant, of a base point free linear system $\left|L\right|$ 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.

Keywords:

• Discriminants
• first jet bundle
• toric varieties

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14M25
• 14N05

Acknowledgments

We would like to thank Javier Pello who suggested us a proof of formula (10(10) $$\begin{array}{c}{c}_{2m}(J_1(D_0+D_{n+1}))=\hfill \\ =\sum _{l=0}^{2m}{(-1)}^l(2m+1-l)\sum _{j=m-l}^{2m-l}\left(\begin{array}{c}2m-l\\ j\end{array}\right)\left(\begin{array}{c}m+1\\ m-j\end{array}\right)\left(\begin{array}{c}m+1\\ l+j-m\end{array}\right)=m+1\hfill \end{array}$$(10) ).

Additional information

Funding

Partially supported by the Spanish government project MTM2015-65968-P.