Algebro-Geometric Method in Commutative Algebra

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We reinterpreted the theory of tight closure in prime characteristic commutative algebra from the viewpoint of singularity theory and birational geometry. Namely, we generalized the concepts of tight closure and F-singularities, gave foundation to the theory thereof, and applied it to problems in commutative algebra and al-gebraic geometry. Our results are summarized as follows:1.Study of F-singularities of Rees algebras : There have been few researches on Rees algebras from a geometric viewpoint, although a Rees algebra is a geometric object in the sense that its "Proj" gives a blow-up. Taking this into account, we studied ring-theoretical and geometric aspects of Rees algebras R(1) associated to an in-primary ideal I of a normal local ring (R,m) in terms of miscellaneous methods such as F-singularities, blow-up and desingularization.2.A generalization of tight closure : We generalized the notion of the tight closure of an ideal in a ring R of characteristic p to those of "D-tight clo … More sure" associated to an effective Q-divisor D on Spec R and of "I-tight closure" associated to an ideal I of R. We established foundation of the theory of I-tight closure and the ideal r(I) defined via I-tight closure, and proved various properties of the ideal -r(I) such as Skoda's theorem, restriction theorem and subadditivity.3.Applications of I-tight closure : We considered the global generation of adjoint bundles K_X+nL of a polarized variety (X, L), as an application of a variant of Skoda's theorem in the canonical module of the section ring of (X,L). In particular, we obtained an alternative proof of K.E.Smith's result on a special case of Fujita's conjecture in characteristic p, assuming that Litself is spanned.We also constructed a characteristic p analog T(‖I.‖) of the asymptotic multiplier ideal associated to a filtration of ideals I.={I_n|n= 1,2,...}. As an application, we reinterpret the result on the uniform behavior of symbolic powers due to Ein-Lazarsfeld-Smith and Hochster-Huneke. Less