Contents Online

# 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

#### 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.