Abstract. In this note we discuss a simple algebraic calculation of the multiplier ideal associated to a monomial ideal in affine n-space. We indicate how this result allows one to compute not only the multiplier ideal but also the log canonical threshold of an ideal in terms of its Newton polygon.

Dedicated to Y.-T. Siu on the occasion of his sixtieth birthday

Abstract. Hara [Ha3] and Smith [S2] independently proved that in a normal Q-Gorenstein ring of characteristic p ≫ 0, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair (R, ∆) of a normal ring R and an effective Q-Weil divisor ∆ on Spec R. As a corollary, we obtain the equivalence of strongly F-regular pairs and klt pairs. 1. introduction Recently it turned out that there exists a relation between multiplier ideals and tight closure. Precisely speaking, it was proven that some algebraic statements established by multiplier ideals could also be understood via tight closure, for example, Briançon-Skoda theorem (see [BS], [HH1], [L]), the problem concerning the growth of symbolic powers of ideals in regular local rings (see [ELS], [HH3]), etc. The purpose of this paper is to give an interpretation of multiplier ideals via tight closure. The theory of tight closure was introduced by Hochster and Huneke [HH1], using the Frobenius map in characteristic p> 0. In this theory, test ideals play a central

Abstract. We prove that certain roots of the Bernstein-Sato polynomial (i.e. b-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith, and D. Varolin. We also prove that certain roots are determined by a filtration on the Milnor cohomology, generalizing a theorem of B. Malgrange in the isolated singularity case. This implies a certain relation with the spectrum which is determined by the Hodge filtration, because the above filtration is related to the pole order filtration. For multiplier ideals we prove an explicit formula in the case of locally conical divisors along a stratification, generalizing a formula of Mustat¸ǎ in the case of hyperplane arrangements. For the b-function of a generic hyperplane arrangement, we determine the multiplicity of −1 and complete a formula of U. Walther.

Abstract. We consider Fano threefolds X with canonical Gorenstein singularities. Under additional assumption that X has at least one noncDV point we prove a sharp bound of the degree: −K3 X ≤ 72. 1.

Abstract. Using a generalization of Malgrange’s formula and a solution of Aomoto’s conjecture due to Esnault, Schechtman and Viehweg, we calculate the Bernstein-Sato polynomial (i.e. b-function) of a hyperplane arrangement with a reduced equation, and show that its roots are greater than −2 and the multiplicity of −1 coincides with the (effective) dimension. As a corollary we get a new proof of Walther’s formula for generic central hyperplane arrangements.

Abstract. In this paper the three-dimensional exceptional strictly log canonical hypersurface singularities are described and the detailed classification of three-dimensional exceptional canonical hypersurface singularities is given under the condition of wellformedness.

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We prove that the set of accumulation points of thresholds in dimension three is equal to the set of thresholds in dimension two, excluding one.

§1 Introduction and statement of results A fundamental problem in classifying varieties is to determine natural subsets whose moduli is bounded. The difficulty of this problem is partially measured by the behaviour of the canonical class of the variety. Three extreme cases are especially of interest: either the canonical class is ample, that is the variety is of