F-rationality of Rees algebras

In this paper, using the notion of the tight integral closure, we will give a criterion for F-rationality of Rees algebras of m-primary ideals in a Cohen-Macaulay local ring. As its application, we prove the following results: (1) In dimension two, if A is F-rational and I is integrally closed, then the Rees algebra R(I) is F-rational. On the other hand, in higher dimensions, we construct many examples of Cohen-Macaulay, normal Rees algebras which are not F-rational. (2) If both A and R(I) are F-rational, then so is the extended Rees algebra R'(I). (3) If R(I) is F-rational and a(G(I)) ≠ - 1, then A is F-rational. On the other hand, using resolution of singularities, we will prove that a two-dimensional rational singularity always admits F-rational Rees algebras. In particular, this theorem gives another way than that devised by Watanabe (1997, J. Pure Appl. Algebra 122, 323-328) to construct counterexamples to the Boutot-type theorem for F-rational rings.