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We prove that the finitistic test ideal τ fg ( R , Δ , a t ) coincides with the big test ideal τ b ( R , Δ , a t ) if the pair ( R , Δ ) is numerically log Q -Gorenstein.

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$., 14 pages

This is a survey article on $F$-singularities and their applications., Comment: 41 pages, to appear in Sugaku Expositions; v2: references added

Let $(S, m)$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $I$ be an ideal of $S$. We prove that if $\text{depth} S/I \ge 3$, then the cohomological dimension $\mathrm{cd}(S, I)$ of $I$ is less than or equal to $n-3$. We also show, under the assumption that $S$ has an algebraically closed residue field of characteristic zero, that if $\text{depth} S/I \ge 4$, then $\mathrm{cd}(S, I) \le n-4$ if and only if the local Picard group of the completion $\widehat{S/I}$ is torsion. We give a number of applications, including sharp bounds on cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre's conditions $(S_i)$., Comment: The Proposition 2.3 of our previous version was not correct as stated. We thank Bhargav Bhatt for pointing this out. It has been fixed in this version. Some minor changes were also added to improve clarity

A conjecture of Hirose, Watanabe, and Yoshida offers a characterization of when a standard graded strongly $F$-regular ring is Gorenstein, in terms of an $F$-pure threshold. We prove this conjecture under the additional hypothesis that the anti-canonical cover of the ring is Noetherian. Moreover, under this hypothesis on the anti-canonical cover, we give a similar criterion for when a normal $F$-pure (resp. log canonical) singularity is quasi-Gorenstein, in terms of an $F$-pure (resp. log canonical) threshold.

We show that every Mori dream space of globally $F$-regular type is of Fano type. As an application, we give a characterization of varieties of Fano type in terms of the singularities of their Cox rings., 22 pages

A singularity in characteristic zero is said to be of dense $F$-pure type if its modulo $p$ reduction is locally Frobenius split for infinitely many $p$. We prove that if $x\in X$ is an isolated log canonical singularity with $\mu (x\in X)\leq 2$ (where the invariant $\mu $ is as defined in Definition 1.4), then it is of dense $F$-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense $F$-pure type in the case of three-dimensional isolated $ \mathbb{Q} $-Gorenstein normal singularities.

Demailly, Ein and Lazarsfeld [3] formulated a subadditivity property of multiplierideals on nonsingular varieties, which states that the multiplier ideal of the prod-uct of two ideal sheaves is contained in the product of their individual multiplierideals. Their formula has many interesting applications in algebraic geometry andcommutative algebra, such as Fujita’s approximation theorem (see [7] and [11, The-orem 10.3.5]) and its local analogue (see [5]), a problem on the growth of symbolicpowers of ideals in regular rings (see [4]), and etc. Later, Takagi [16] and Eisenstein[6] generalized their formula to the case of Q-Gorenstein varieties, that is, the casewhen ∆ = 0 in the above definition of multiplier ideals. In this article, we studya further generalization to the case of log pairs, when the importance of multiplierideals is particularly highlighted. The following is our main result.Theorem (Theorems 2.3 and 3.5). Let Xbe a normal variety over an algebraicallyclosed field of characteristic zero and ∆ be an effective Q-divisor on X such thatr(K

We prove that the $F$-jumping numbers of the test ideal $\tau(X; \Delta, \ba^t)$ are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\Delta$ is $\bQ$-Cartier of index not divisible $p$, and either $X$ is essentially of finite type over a field or the sheaf of ideals $\ba$ is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero., Comment: 29 pages, minor changes, to appear in Mathematische Annalen

Smith and Van den Bergh introduced the notion of finite F-representation type as a characteristic $p$ analogue of the notion of finite representation type. In this paper, we prove two finiteness properties of rings with finite F-representation type. The first property states that if $R=\bigoplus_{n \ge 0}R_n$ is a Noetherian graded ring with finite (graded) F-representation type, then for every non-zerodivisor $x \in R$, $R_x$ is generated by $1/x$ as a $D_{R}$-module. The second one states that if $R$ is a Gorenstein ring with finite F-representation type, then $H_I^n(R)$ has only finitely many associated primes for any ideal $I$ of $R$ and any integer $n$. We also include a result on the discreteness of F-jumping exponents of ideals of rings with finite (graded) F-representation type as an appendix., Comment: 19 pages; v.2: minor changes, to appear in Math. Res. Lett

An $F$-nilpotent local ring is a local ring $(R, \mathfrak{m})$ of prime characteristic defined by the nilpotence of the Frobenius action on its local cohomology modules $H^i_{\mathfrak{m}}(R)$. A singularity in characteristic zero is said to be of $F$-nilpotent type if its modulo $p$ reduction is $F$-nilpotent for almost all $p$. In this paper, we give a Hodge-theoretic interpretation of three-dimensional normal isolated singularities of $F$-nilpotent type. In the graded case, this yields a characterization of these singularities in terms of divisor class groups and Brauer groups., Comment: 18 pages; v2: several typos fixed

We prove a generalization of Demailly-Ein-Lazarsfeld's subadditivity formula and Mustata's summation formula for multiplier ideals to the case of singular varieties, using characteristic $p$ methods. As an application of our formula, we improve Hochster-Huneke's result on the growth of symbolic powers of ideals in singular affine algebras., 16 pages, AMS-LaTeX; v.2: Some minor changes

Demailly, Ein and Lazarsfeld \cite{DEL} proved the subadditivity theorem for multiplier ideals, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals, on non-singular varieties. We prove that, in two-dimensional case, the subadditivity theorem holds on log-terminal singularities. However, in higher dimensional case, we have several counter-examples. We consider the subadditivity theorem for monomial ideals on toric rings, and construct a counter-example on a three-dimensional toric ring., Comment: 12 pages, AMS-LaTeX; v.2: minor changes, to appear in Trans. Amer. Math. Soc

The test ideal $\tau(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $\tau(\a^t)$ associated to a given ideal $\a$ with rational exponent $t \ge 0$. We first prove a key lemma of this paper, which gives a characterization of the ideal $\tau(\a^t)$. As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal $\tau(R)$. Moreover, we prove an analog of so-called Skoda's theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the "modified Brian\c{c}on--Skoda theorem.", Comment: 11 pages, AMS-LaTeX; v.2: minor changes, to appear in Nagoya Math. J

We show that the reduction to positive characteristic of the multiplier ideal in the sense of de Fernex and Hacon agrees with the test ideal for infinitely many primes, assuming that the variety is numerically Q-Gorenstein. It follows, in particular, that this reduction property holds in dimension 2 for all normal surfaces., 11 pages; v2: minor changes, to appear in Bull. London Math. Soc

We prove that normal projective surfaces of dense globally $F$-split type (resp. globally $F$-regular type) are of Calabi-Yau type (resp. Fano type)., Comment: 18 pages, v2: expanded explanations, typos corrected

This paper presents three results on F-singularities. First, we give a new proof of Eisenstein's restriction theorem for adjoint ideal sheaves, using the theory of F-singularities. Second, we show that a conjecture of Musta\c{t}\u{a} and Srinivas implies a conjectural correspondence of F-purity and log canonicity. Finally, we prove this correspondence when the defining equations of the variety are very general., Comment: 23 pages, v2: Introduction revised, Section 2 expanded (Example 2.9 and Remarks 2.6, 2.8 and 2.12 added), new references added, no other substantial changes, v3: an error in the proof of Theorem 2.10 of the previous version corrected, v4: an error in Example 2.10 corrected, other minor changes, to appear in Algebra Number Theory, v5: an error in the proof of Theorem 2.11 fixed

The $F$-threshold $c^J(\a)$ of an ideal $\a$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We study a conjecture formulated in an earlier paper \cite{HMTW} by the same authors together with M. Musta\c{t}\u{a}, which bounds $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\a$ and $J$ are generated by homogeneous systems of parameters in a Noetherian graded $k$-algebra. We also prove a similar inequality involving, instead of the $F$-threshold, the jumping number for the generalized parameter test submodules introduced in \cite{ST}., Comment: 19 pages; v.2: a new section added, treating a comparison of F-thresholds and F-jumping numbers

We prove that the log canonical thresholds of a large class of binomial ideals, such as complete intersection binomial ideals and the defining ideals of space monomial curves, are computable by linear programming., 16 pages, 1 figure; v2: expanded explanations, typos corrected; v3: minor changes, to appear in Manuscripta Math

The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect the containment in the integral closure or the tight closure of a parameter ideal using F-thresholds. We formulate a conjecture bounding $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals, and $J$ is generated by a system of parameters. We prove the conjecture when $J$ is a monomial ideal in a polynomial ring, and also when $\a$ and $J$ are generated by homogeneous systems of parameters in a Cohen-Macaulay graded $k$-algebra., Comment: 22 pages; v.2: minor changes, to appear in Michigan Math. J

Hochster and Huneke proved in \cite{HH5} fine behaviors of symbolic powers of ideals in regular rings, using the theory of tight closure. In this paper, we use generalized test ideals, which are a characteristic $p$ analogue of multiplier ideals, to give a slight generalization of Hochster-Huneke's results., Comment: 13 pages; v.2: minor changes, to appear in Michigan Math. J

We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair $(X,\Delta)$ is plt if and only if the modulo $p$ reduction of $(X,\Delta)$ is divisorially F-regular for all large $p \gg 0$. Here, divisorially F-regular pairs are a class of singularities in positive characteristic introduced by Hara and Watanabe in terms of Frobenius splitting., Comment: 22 pages. A lot of changes. Some errors and many typos fixed

We generalize the notions of F-regular and F-pure rings to pairs $(R,\a^t)$ of rings $R$ and ideals $\a \subset R$ with real exponent $t > 0$, and investigate these properties. These ``F-singularities of pairs'' correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove Inversion of Adjunction of arbitrary codimension, which states that for a pair $(X,Y)$ of a smooth variety $X$ and a closed subscheme $Y \subsetneq X$, if the restriction $(Z, Y|_Z)$ to a normal $\Q$-Gorenstein closed subvariety $Z \subsetneq X$ is klt (resp. lc), then the pair $(X,Y+Z)$ is plt (resp. lc) near $Z$., 21 pages, AMS-LaTeX; v.2: minor changes, to appear in Invent. Math

Using the Frobenius map, we introduce a new invariant for a pair $(R,\a)$ of a ring $R$ and an ideal $\a \subset R$, which we call the F-pure threshold $\mathrm{c}(\a)$ of $\a$, and study its properties. We see that the F-pure threshold characterizes several ring theoretic properties. By virtue of Hara and Yoshida's result, the F-pure threshold $\mathrm{c}(\a)$ in characteristic zero corresponds to the log canonical threshold $\mathrm{lc}(\a)$ which is an important invariant in birational geometry. Using the F-pure threshold, we prove some ring theoretic properties of three-dimensional terminal singularities of characteristic zero. Also, in fixed prime characteristic, we establish several properties of F-pure threshold similar to those of the log canonical threshold with quite simple proofs., Comment: 19 pages; v.2: minor changes, to appear in J. Algebra

Hara and Smith independently proved that in a normal $\mQ$-Gorenstein ring of characteristic $p \gg 0$, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair $(R, \Delta)$ of a normal ring $R$ and an effective $\mQ$-Weil divisor $\Delta$ on $\Spec R$. As a corollary, we obtain the equivalence of strongly F-regular pairs and klt pairs., Comment: 21 pages in AMS LaTeX, to appear in Journal of Algebraic Geometry

A Stefan’s problem in a finite domain may be given an approximate analytical solution. An example is shown with constant boundary and initial conditions. The solution is initially that of a semi-infinite domain, transits through infinitely many intermediate stage solutions, and finally becomes stationary. The solution is exact in the initial stage and also at the steady final stage, but approximate at the intermediate stages.

Presented in this paper is the theory of the steady in-plane deformation, obeying the Coulomb yield criterion, of plastic soils whose strain rate and stress principal directions are noncoaxial. The constitutive equations including an unknown noncoaxial angle are derived by use of the geometry of the Mohr circle and the theory of characteristic lines. A boundary value problem is solved by assigning to the non coaxial angle a set of such values that enable us to accommodate the presupposed type of flow satisfying the given boundary conditions in a given domain. The plastic material regulated by the Coulomb yield criterion in in-plane deformation is, therefore, a singular material whose constitutive equations are not constant with material but are variable with flow conditions.

A mechanism of ice lens formation is presented on the assumption that its main cause is the simultaneous flow of heat and water. The differential equations thus formulated are solved approximately by the use of a generalization of Goodman's integral method. The result is found to be not completely satisfactory when compared with an experiment. The following progress has however been made: (1) The existence of a solution to the differential equations of ice lens formation is demonstrated. (2) The solution exists for a very narrow range of initial water content determined by the amount of surcharge on the ice lens. The narrowness of the range probably accounts for the observed intermittence of lenses. (3) The Goodman technique or any generalization of it cannot yield a satisfactory solution of the problem. (4) A simplification of Portnov's method has been found. (5) The possibility of obtaining an exact solution for a short initial time interval arises when the independent variables x and t are reduced to x/(t)½ and (t)½, where x is the vertical coordinate and t is the time. (6) With development of the mathematical method, improvements in the assumed mechanism, and refinements in the measurements of soil properties, we may be able to completely formulate frost heaving and thereby solve a long-standing problem.