Cartier modules on toric varieties

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 ${\varphi }_{\Delta }:F_{\ast }^e{O}_X\to O_X$ be the toric map corresponding to $\Delta$. We identify all ideals $I$ of $O_X$ with ${\varphi }_{\Delta }(F_{\ast }^{e}I)=I$ 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 ${\varphi }_{\Delta }$ and $\ba$; this answers a question by Manuel Blickle in the toric setting.

21 pages, minor changes, example 3.10 added. To appear in Transactions of the American Mathematical Society

References in corpus (2)

• A survey of test ideals
• Basic properties of log canonical centers

Cited by in corpus (2)

• On the three dimensional minimal model program in positive characteristic
• Compatible ideals in Gorenstein rings