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The wild McKay correspondence, a variant of the McKay correspondence in positive characteristics, shows that stringy motives of quotient varieties equal some motivic integrals on the moduli space of the Galois covers of a formal disk. In this paper, we determine when the integral converges for the case of cyclic groups of prime power order. As an application, we give a criterion for the quotient variety being canonical or log canonical. [ABSTRACT FROM AUTHOR]

We show that each direct summand of the associated graded module of the test module filtration τ (M , f λ) λ ≥ 0 admits a natural Cartier structure. If λ is an F-jumping number, then this Cartier structure is nilpotent on τ (M , f λ − ε) / τ (M , f λ) if and only if the denominator of λ is divisible by p. We also show that these Cartier structures coincide with certain Cartier structures that are obtained by considering certain D -modules associated to M that were used to construct Bernstein-Sato polynomials. Moreover, we point out that the zeros of the Bernstein-Sato polynomial b M , f attached to an F-regular Cartier module correspond to its F-jumping numbers. This generalizes [6, Theorem 5.4] where a stronger version of F-regularity was used. Finally, we develop a basic theory of non-F-pure modules and prove a weaker connection between Bernstein-Sato polynomials b M , f and Cartier modules (M , κ) for which Mf is F-regular and certain jumping numbers attached to M. [ABSTRACT FROM AUTHOR]

For an affine toric variety Spec (A) , we give a convex geometric interpretation of the Gerstenhaber product HH 2 (A) × HH 2 (A) → HH 3 (A) between the Hochschild cohomology groups. In particular, our construction also generalizes the known results about the cup product T A 1 × T A 1 → T A 2 . [ABSTRACT FROM AUTHOR]

Given a polynomial f with coefficients in a field of prime characteristic p, it is known that there exists a differential operator that raises 1 / f to its pth power. We first discuss a relation between the "level" of this differential operator and the notion of "stratification" in the case of hyperelliptic curves. Next, we extend the notion of level to that of a pair of polynomials. We prove some basic properties and we compute this level in certain special cases. In particular, we present examples of polynomials g and f such that there is no differential operator raising g/f to its pth power. [ABSTRACT FROM AUTHOR]

We construct a differential graded Lie algebra g controlling the Poisson deformations of an affine Poisson variety. We analyze g in the case of affine Gorenstein toric Poisson varieties. Moreover, explicit description of the second and third Hochschild cohomology groups is given for three-dimensional affine Gorenstein toric varieties. [ABSTRACT FROM AUTHOR]

The Hodge spectrum is an important analytic invariant of singularities encoding the Hodge filtration and the monodromy of the Milnor fiber. However, explicit formulas exist in only a few cases. In this article, the main result is a combinatorial formula for the Hodge spectrum of any homogeneous polynomials in three variables whose zero locus is a projective curve arrangement having only ordinary multiple points. [ABSTRACT FROM AUTHOR]

We study the singularities of the isospectral Hilbert scheme Bn of n points over a smooth algebraic surface and we prove that they are canonical if , log-canonical if and not log-canonical if. We describe as well two explicit log-resolutions of B3, one crepant and the other -equivariant. [ABSTRACT FROM AUTHOR]

We compute Denef and Loeser's motivic zeta function associated to a monomial ideal on an affine toric variety, generalizing a result of Howald, Mustaţă, and Yuen. We also investigate the relation between the poles of the motivic zeta function and the roots of their corresponding Bernstein–Sato polynomial defined by the first author and Matusevich. [ABSTRACT FROM AUTHOR]

In [6], de Fernex and Hacon started the study of singularities on non-ℚ-Gorenstein varieties using pullbacks of Weil divisors. In [4], the author of this article and Urbinati introduce a new class of singularities, calledlog terminal+, or simplylt+, which they prove is rather well behaved. In this article we will continue the study of lt+singularities, and we will show that they can be detected by a multiplier ideal, that they satisfy a Bertini type result, inversion of adjunction, and small deformation invariance, and that they are naturally related to rational singularities. Finally, we will provide a list of examples (all of them with lt+singularities) of the pathologies that can occur in the study of non-ℚ-Gorenstein singularities. [ABSTRACT FROM AUTHOR]

Suppose that π:Y → Xis a finite map of normal varieties over a perfect field of characteristicp > 0. Previous work of the authors gave a criterion for when Frobenius splittings onX(or more generally anyp−e-linear map) extend toY. In this paper we give an alternate and highly explicit proof of this criterion (checking term by term) when π is tamely ramified in codimension 1. Some additional examples are also explored. [ABSTRACT FROM AUTHOR]

We generalize Zariski's Perron transforms to obtain a proof of local uniformization of an arbitrary singular variety of characteristic zero along a valuation of arbitrary rank. Furthermore, we resolve the centers of all the composite valuations as well as the prime ideals of infinite value. [ABSTRACT FROM AUTHOR]

We provide some examples of simple small singularities of higher dimensional algebraic varieties. One of them is anE6type singularityw2 − z3 + xy3 − 3x2yz − x5 + xzw − x4y = 0 in ℂ4. We also treat small contractions of curves with higher genera whose normal bundles are not negative. [ABSTRACT FROM AUTHOR]

In this short note we show that for any pair of positive integers (d,n) withn > 2 andd > 1 orn = 2 andd > 4, there always exist projective varietiesX ⊆ ℙNof dimensionnand degreedand an integers0such that Hilbs(X) is reducible for alls ≥ s0.Xwill be a projective cone in ℙNover an arbitrary projective varietyY ⊆ ℙN−1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points. [ABSTRACT FROM AUTHOR]

We construct a class of projective rational varieties X of any dimension m ≥ 1, which are smooth except at a point O, with the projective space ℙ m as normalization, having smooth branches, and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sufficient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal. [ABSTRACT FROM PUBLISHER]

We show that for a class of modules over shod algebras, including the canonical tilting modules, the closures of the corresponding orbits in module varieties are regular in codimension one. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective variety of dimension r and π:X → ℙm a generic projection with r + 1 ≤ m ≤ 2r. It is shown that, at any point on X' = π(X) of multiplicity μ, off a closed subset of the triple locus of codimension four, the depth of the local ring is equal to r - (μ - 1)(m - r - 1). This leads to some improvements on the affirmation of a conjecture of Andreotti-Bombieri-Holm on the weak normality of X' and a conjecture of Piene on the weak normality of Sing(X'). [ABSTRACT FROM AUTHOR]

We consider the multiplier ideals of the ideal of a reduced union of lines through the origin in 3. For general arrangements of lines, we calculate the multiplier ideals. [ABSTRACT FROM AUTHOR]

The sequence of centers and multiplicities for any valuation of the local ring of a point p in an n-dimensional non-singular variety is defined. A Noether formula for valuations is obtained, which leads to some proximity relations between the multiplicities of the valuation in its centers. We will consider a sequence of irreducible subschemes infinitely near p, each one in the first neighborhood of the preceding one, and with the condition that infinitely many of them are free. Then, a valuation with those centers and multiplicities exists which, unlike the plane case, is not unique. To show this, an example of a space valuation, built as the lifting of a plane valuation, is provided. [ABSTRACT FROM AUTHOR]

We give an algorithm that describes the singularity of all but finitely-many germs in a pencil generated by two arbitrary germs of plane curve. [ABSTRACT FROM AUTHOR]