BLOW-UP RINGS AND F-RATIONALITY

In this paper, we prove some sufficient conditions for Cohen–Macaulay normal Rees algebras to be F-rational. Let (R,𝔪) be a Gorenstein normal local domain of dimension d≥2 and of characteristic p>0. Let I be an ideal generated by a system of parameters. Our first set of results give conditions on the test ideals τ(In), n≥1 which would imply that the normalization of the Rees algebra R[It] is F-rational. Another sufficient condition is that the socle of HG¯+d(G¯) (where 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 Proj(R[𝔪t]) is F-rational, then R[𝔪t] is F-rational.