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Cartier modules on toric varieties

Authors: Jen-Chieh Hsiao, Karl Schwede and Wenliang Zhang
Journal: Trans. Amer. Math. Soc. 366 (2014), 1773-1795
MSC (2010): Primary 14M25, 13A35, 14F18, 14B05
Published electronically: November 25, 2013
MathSciNet review: 3152712
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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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Additional Information

Jen-Chieh Hsiao
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Address at time of publication: Department of Mathematics, National Cheng Kung University, No. 1 University Road, Tainan 701, Taiwan R.O.C.

Karl Schwede
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Wenliang Zhang
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588

Received by editor(s): August 18, 2011
Received by editor(s) in revised form: April 10, 2012
Published electronically: November 25, 2013
Additional Notes: The first author was partially supported by the NSF grant DMS #0901123
This research was initiated at the Commutative Algebra MRC held in June 2010. Support for this meeting was provided by the NSF and AMS
The second author was supported by an NSF postdoctoral fellowship and also by NSF grant DMS #1064485
The third author was partially supported by the NSF grant DMS #1068946.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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