Mathematical Research Letters

Volume 19 (2012)

Number 2

$F$-purity of hypersurfaces

Pages: 389 – 401

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n2.a11

Author

Daniel J. Hernández (Department of Mathematics, University of Minnesota, Twin Cities, 127 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, U.S.A.)

Abstract

Motivated by connections with birational geometry, the theory of $F$-purity for rings ofpositive characteristic may be extended to a theory of$F$-purity for “pairs”~\cite{HW2002}. Given an element$f$ of an $F$-pure ring of positive characteristic, thisextension allows us to define the $F$-pure threshold of$f$, denoted $\fpt{f}$. This invariant measures thesingularities of $f$, and may be thought of as a positivecharacteristic analog of the log canonical threshold, aninvariant that typically appears in the study ofsingularities of hypersurfaces over $\mathbb{C}$. In thisnote, we study $F$-purity of pairs, and show (as is thecase with log canonicity) that $F$-purityis preserved at the $F$-pure threshold. We alsocharacterize when $F$-purity is equivalent to \emph{sharp}$F$-purity, an alternate notion of purity for pairsintroduced in~\cite{Schwede2008}. These results on purityat the threshold generalize results appearing in\cite{Hara2006,Schwede2008}, and were expected to hold bymany experts in the field. We conclude by extendingresults in~\cite{BMS2009} on the set of all $F$-purethresholds to the most general\break setting.

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