Irreducible components of Toric Complete Intersections

TL;DR

This paper develops a recursive formula to determine the number of irreducible components in complete intersections of general divisors on any toric variety, extending previous explicit formulas for the case of algebraic tori.

Contribution

It introduces a recursive method for computing irreducible components of equivariant linear systems on arbitrary toric varieties, generalizing earlier explicit formulas.

Findings

Derived a recursive formula for irreducible components on toric varieties.

Extended Khovanskii's explicit formula from algebraic tori to general toric varieties.

Provided a theoretical framework for analyzing equivariant linear systems.

Abstract

An equivariant linear system on a toric variety is a linear system invariant under the torus action. We study the number of irreducible components of the complete intersection of general divisors from a fixed collection of equivariant linear system on a toric variety $X$. An explicit formula for the number of components was obtained by Khovanskii in 2016 for the case $X = T^n$ over $\mathbb C$ and generalized to an algebraically closed field of arbitrary characteristic the author in 2024. Building on these results, we give a recursive formula for an arbitrary toric variety.