Cartier modules on toric varieties

TL;DR

The paper characterizes certain fixed ideals on affine toric varieties in positive characteristic using combinatorial and log resolution methods, extending to characteristic zero and answering a question by Blickle.

Contribution

It provides a combinatorial and geometric description of ideals fixed by Cartier operators on toric varieties, including those fixed by generated Cartier algebras, in both positive characteristic and characteristic zero.

Findings

Identifies ideals fixed by a toric Cartier map combinatorially.

Describes fixed ideals in terms of log resolutions.

Answers a question by Manuel Blickle regarding Cartier algebra fixed ideals.

Abstract

Assume that $X$ is an affine toric variety of characteristic $p > 0$. Let $\Delta$ be an effective toric $Q$-divisor such that $K_X+\Delta$ is $Q$-Cartier with index not divisible by $p$ and let $\phi_{\Delta}:F^e_* O_X \to O_X$ be the toric map corresponding to $\Delta$. We identify all ideals $I$ of $O_X$ with $\phi_{\Delta}(F^e_* I)=I$ combinatorially and also in terms of a log resolution (giving us a version of these ideals which can be defined in characteristic zero). Moreover, given a toric ideal $\ba$, we identify all ideals $I$ fixed by the Cartier algebra generated by $\phi_{\Delta}$ and $\ba$; this answers a question by Manuel Blickle in the toric setting.