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This study examines the finite $F$-representation type (abbr. FFRT) property of a two-dimensional normal graded ring $R$ in characteristic $p>0$, using notions from the theory of algebraic stacks. Given a graded ring $R$, we consider an orbifold curve $\mathfrak C$, which is a root stack over the smooth curve $C=\text{Proj} R$, such that $R$ is the section ring associated with a line bundle $L$ on $\mathfrak C$. The FFRT property of $R$ is then rephrased with respect to the Frobenius push-forwards $F^e_*(L^i)$ on the orbifold curve $\mathfrak C$. As a result, we see that if the singularity of $R$ is not log terminal, then $R$ has FFRT only in exceptional cases where the characteristic $p$ divides a weight of $\mathfrak C$., 25 pages, exposition on stacks added, to appear in Adv. Math

We extend Kawachi's invariant for reduced points on a normal suface to an invariant defined for possibly non-reduced zero-dimensional closed subschemes (or fat points) on a surface with rational singularities. We give an upper bound of this invariant in terms of the length of the fat point and the discrepancy of the minimal resolution. As an application we prove a Reider-type theorem for k-very ampleness on surfaces with rational singularities.

The notions of F -rational and F -regular rings are defined via tight closure, which is a closure operation for ideals in a commutative ring of positive characteristic. The geometric significance of these notions has persisted, and K. E. Smith proved that F -rational rings have rational singularities. We now ask about the converse implication. The answer to this question is yes and no. For a fixed positive characteristic, there is a rational singularity which is not F -rational, so the answer is no. In this paper, however, we aim to show that the answer is yes in the following sense: If a ring of characteristic zero has rational singularity, then its modulo p reduction is F -rational for almost all characteristic p . This result leads us to the correspondence of F -regular rings and log terminal singularities.

LetR be a two-dimensional normal graded ring over a field of characteristicp>0. We want to describe the tight closure of (O) in the local cohomology moduleH R+ 2 (R) using the graded module structure ofH R+ 2 (R). For this purpose we explore the condition that the Frobenius mapF: [H R+ 2 (R)]n→[H R+ 2 (R)]pninduced on graded pieces ofH R+ 2 (R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisorD onX=Proj (R) such thatR=R (X, D)= ⊕ n≥0H0(X O X (n D)). Then the above map is identified with the induced Frobenius on the cohomology groups Our interest is the casen0 is defined via tight closure and is expected to characterize rational singularities. We ask if a modulop reduction of a rational signularity in characteristic 0 isF-rational forp≫0. Our result answers to this question affirmatively and also sheds light to behavior ofF-rationality in smallp.

We study F-blowups of non-F-regular normal surface singularities. Especially the cases of rational double points and simple elliptic singularities are treated in detail., 25 pages, v2: minor revision

We introduce a new variant of tight closure associated to any fixed ideal $\a$, which we call $\a$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $\tau(\a)$ of all $\a$-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal $\tau(\a)$ and the multiplier ideal associated to $\a$ (or, the adjoint of $\a$ in Lipman's sense) in normal $\Q$-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prime characteristic, we establish some properties of $\tau(\a)$ similar to those of multiplier ideals (e.g., a Brian\c{c}on-Skoda type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal $\tau(\a)$ and the F-rationality of Rees algebras., Comment: about 35 pages, to appear in Trans. Amer. Math. Soc

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. Algebra122, 323–328) to construct counterexamples to the Boutot-type theorem for F-rational rings.