Blow-up rings and $F$-rationality

Abstract

In this paper, we prove some sufficient conditions for Cohen-Macaulay normal Rees algebras to be $F$-rational. Let $(R,\mathfrak{m})$ be a Gorenstein normal local domain of dimension $d\ge 2$ and of characteristic $p>0$. Let $I$ be a $\mathfrak{m}$-primary ideal. Our first set of results give conditions on the test ideals $\tau (I^n)$, $n\ge 1$ which would imply that the normalization of the Rees algebra $R[It]$ is $F$-rational. Another sufficient condition is that the socle of ${\mathrm{H}}_{{\overline{G}}_{+}}^d(\overline{G})$ (where $\overline{G}$ is the associated graded ring for the integral closure filtration) is entirely in degree $-1$, if $R$ is $F$-rational (but not necessarily Gorenstein). Then we show that if $R$ is a hypersurface of degree $2$ or is three-dimensional and $F$-rational, and $\mathrm{Proj}(R[\mathfrak{m}t])$ is $F$-rational, then $R[\mathfrak{m}t]$ is $F$-rational.