Cartier modules on toric varieties

Hsiao, Jen-Chieh; Schwede, Karl; Zhang, Wenliang

doi:10.1090/S0002-9947-2013-05856-4

Cartier modules on toric varieties
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by Jen-Chieh Hsiao, Karl Schwede and Wenliang Zhang PDF

Trans. Amer. Math. Soc. 366 (2014), 1773-1795 Request permission

Abstract:

Assume that $X$ is an affine toric variety of characteristic $p > 0$. Let $\Delta$ be an effective toric $\mathbb {Q}$-divisor such that $K_X+\Delta$ is $\mathbb {Q}$-Cartier with index not divisible by $p$ and let $\phi _{\Delta }:F^e_*\mathscr {O}_X\to \mathscr {O}_X$ be the toric map corresponding to $\Delta$. We identify all ideals $I$ of $\mathscr {O}_X$ 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 $\mathfrak {a}$, we identify all ideals $I$ fixed by the Cartier algebra generated by $\phi _{\Delta }$ and $\mathfrak {a}$; this answers a question by Manuel Blickle in the toric setting.

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