Search

Let A be a Cohen–Macaulay normal domain. A non-commutative crepant resolution (NCCR) of A is an A -algebra Γ of the form Γ = End A ( M ) , where M is a reflexive A -module, Γ is maximal Cohen–Macaulay as an A -module and gldim ( Γ ) P = dim ⁡ A P for all primes P of A . We give bountiful examples of equi-characteristic Cohen–Macaulay normal local domains and mixed characteristic Cohen–Macaulay normal local domains having NCCR. We also give plentiful examples of affine Cohen–Macaulay normal domains having NCCR. [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]

In this article, we prove that the defect of all simple extension of valued field is the product of the effective degrees of the complete set of key polynomials associated with. As a consequence, we obtain a local uniformization theorem for valuations of rank 1 centered on an equicharacteristic quasi-excellent local domain satisfying some inductive assumptions of lack of defect. [ABSTRACT FROM AUTHOR]

For a generic f ∈ C ∞ ( R 4 , R 4 ) there is a discrete set Σ 2 ( D f ) of critical points of rank two, and there is an integer index I p ( D f ) associated to any p ∈ Σ 2 ( D f ) . We show how to compute ∑ I p ( D f ) , where p ∈ Σ 2 ( D f ) , in the case where f is a polynomial mapping. [ABSTRACT FROM AUTHOR]

The monodromy conjecture predicts that the poles of the topological zeta function and related zeta functions associated to a polynomial [ABSTRACT FROM AUTHOR]

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$. To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity. [ABSTRACT FROM PUBLISHER]

Given an analytic function germ f: ( X, 0) → C on an isolated determinantal singularity or on a reduced curve, we present formulas relating the local Euler obstruction of f to the vanishing Euler characteristic of the fiber X ∩ f (0) and to the Milnor number of f. Restricting ourselves to the case where X is a complete intersection, we obtain an easy way to calculate the local Euler obstruction of f as the difference between the dimension of two algebras. [ABSTRACT FROM AUTHOR]

This paper deals with computing the global dimension of endomorphism rings of maximal Cohen–Macaulay (=MCM) modules over commutative rings. Several examples are computed. In particular, we determine the global spectra, that is, the sets of all possible finite global dimensions of endomorphism rings of MCM-modules, of the curve singularities of type A n for all n , D n for n ≤ 13 and E 6 , 7 , 8 and compute the global dimensions of Leuschke's normalization chains for all ADE curves, as announced in [12] . Moreover, we determine the centre of an endomorphism ring of a MCM-module over any curve singularity of finite MCM-type. In general, we describe a method for the computation of the global dimension of an endomorphism ring End R M , where R is a Henselian local ring, using add ( M ) -approximations. When M ≠ 0 is a MCM-module over R and R is Henselian local of Krull dimension ≤2 with a canonical module and of finite MCM-type, we use Auslander–Reiten theory and Iyama's ladder method to explicitly construct these approximations. [ABSTRACT FROM AUTHOR]

We develop a theory of 'ad hoc' Chern characters for twisted matrix factorizations associated to a scheme X, a line bundle $$\mathcal {L}$$ , and a regular global section $$W \in \Gamma (X, \mathcal {L})$$ . As an application, we establish the vanishing, in certain cases, of $$h_c^R(M,N)$$ , the higher Herbrand difference, and, $$\eta _c^R(M,N)$$ , the higher codimensional analogue of Hochster's theta pairing, where R is a complete intersection of codimension c with isolated singularities and M and N are finitely generated R-modules. Specifically, we prove such vanishing if $$R = Q/(f_1, \dots , f_c)$$ has only isolated singularities, Q is a smooth k-algebra, k is a field of characteristic 0, the $$f_i$$ 's form a regular sequence, and $$c \ge 2$$ . Such vanishing was previously established in the general characteristic, but graded, setting in Moore et al. (Math Z 273(3-4):907-920, 2013). [ABSTRACT FROM AUTHOR]

Let ( R , m ) be a Noetherian local ring of dimension d > 0 . Let I • = { I n } n ∈ N be a graded family of m -primary ideals in R . We examine how far off from a polynomial can the length function ℓ R ( R / I n ) be asymptotically. More specifically, we show that there exists a constant γ > 0 such that for all n ≥ 0 , ℓ R ( R / I n + 1 ) − ℓ R ( R / I n ) < γ n d − 1 . [ABSTRACT FROM AUTHOR]

The periodic cyclic homology of any proper dg category comes equipped with a canonical pairing. We show that in the case of the dg category of matrix factorizations of an isolated singularity the canonical pairing can be identified with Kyoji Saito's higher residue pairing on the twisted de Rham cohomology of the singularity. [ABSTRACT FROM AUTHOR]

Let G be a semisimple algebraic group defined over $$\mathbb {Q}_p$$ , and let $$\Gamma $$ be a compact open subgroup of $$G(\mathbb {Q}_p)$$ . We relate the asymptotic representation theory of $$\Gamma $$ and the singularities of the moduli space of G-local systems on a smooth projective curve, proving new theorems about both: For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities. [ABSTRACT FROM AUTHOR]

Based on Tiankai Liu's PhD thesis [], we give a complete classification of local topological types of singularities with two Newton pairs on rational unicuspidal complex projective plane curves. [ABSTRACT FROM AUTHOR]

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz regular complex analytic set, i.e., any complex analytic set which is locally bi-Lipschitz homeomorphic to an Euclidean ball must be smooth. Finally, we give an alternative proof of S. Koike and L. Paunescu's result about the bi-Lipschitz invariance of directional dimensions of subanalytic sets. [ABSTRACT FROM AUTHOR]

Let a ⊂ O X be a coherent ideal sheaf on a normal complex variety X , and let c ≥ 0 be a real number. De Fernex and Hacon associated a multiplier ideal sheaf to the pair ( X , a c ) which coincides with the usual notion whenever the canonical divisor K X is Q -Cartier. We investigate the properties of the jumping numbers associated to these multiplier ideals. We show that the set of jumping numbers of a pair is unbounded, countable and satisfies a certain periodicity property. We then prove that the jumping numbers form a discrete set of real numbers if the locus where K X fails to be Q -Cartier is zero-dimensional. It follows that discreteness holds whenever X is a threefold with rational singularities. Furthermore, we show that the jumping numbers are rational and discrete if one removes from X a closed subset W ⊂ X of codimension at least three, which does not depend on a . We also obtain that outside of W , the multiplier ideal reduces to the test ideal modulo sufficiently large primes p ≫ 0 . [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]

To a graded finite-rank matrix factorisation of the difference of two homogeneous potentials one can assign two numerical invariants, the left and right quantum dimensions. The existence of such a matrix factorisation with non-zero quantum dimensions defines an equivalence relation between potentials, giving rise to non-obvious equivalences of categories. Restricted to ADE singularities, the resulting equivalence classes of potentials are those of type { A d − 1 } for d odd, { A d − 1 , D d / 2 + 1 } for d even but not in { 12 , 18 , 30 } , and { A 11 , D 7 , E 6 } , { A 17 , D 10 , E 7 } and { A 29 , D 16 , E 8 } . This is the result expected from two-dimensional rational conformal field theory, and it directly leads to new descriptions of and relations between the associated (derived) categories of matrix factorisations and Dynkin quiver representations. [ABSTRACT FROM AUTHOR]

This paper addresses a very classical topic that goes back at least to Plücker: how to understand a plane curve singularity using its polar curves. Here, we explicitly construct the singular points of a plane curve singularity directly from the weighted cluster of base points of its polars. In particular, we determine the equisingularity class (or topological equivalence class) of a germ of plane curve from the equisingularity class of generic polars and combinatorial data about the non-singular points shared by them. [ABSTRACT FROM AUTHOR]

Let V ⊂ P n ( F ‾ q ) be a complete intersection defined over a finite field F q of dimension r and singular locus of dimension at most 0 ≤ s ≤ r − 2 . We obtain an explicit version of the Hooley–Katz estimate | | V ( F q ) | − p r | = O ( q ( r + s + 1 ) / 2 ) , where | V ( F q ) | denotes the number of F q -rational points of V and p r : = | P r ( F q ) | . Our estimate improves all the previous estimates in several important cases. Our approach relies on tools of classical algebraic geometry. A crucial ingredient is a new effective version of the Bertini smoothness theorem, namely an explicit upper bound of the degree of a proper Zariski closed subset of ( P n ) s + 1 ( F ‾ q ) which contains all the singular linear sections of V of codimension s + 1 . [ABSTRACT FROM AUTHOR]

It is well known that the main approaches of the analytical solving of the elasticity mixed plane problems for a semi-strip are based on the different representations of the equilibrium equations' solutions: the representations through the harmonic and by harmonic functions, through the stress function, Fadle-Papkovich functions and so on. The main shortcoming of these approaches is connected with the fact that to obtain the expression for the real mechanical characteristics, one should execute additional operations, not always simple ones. The approach that is proposed in this paper allows the direct solution of the equilibrium equations. With the help of the matrix integral transformation method applied directly to the equilibrium equations, the initial boundary problem is reduced to a vector boundary problem in the transformation's domain. The use of matrix differential calculations and Green's matrix function leads to the exact vector solution of the problem. Green's matrix function is constructed in the form of a bilinear representation which simplifies the calculations. The method is demonstrated by the solving of the thermoelastic problem for the semi-strip. The zones and conditions of the strain stress occurrence on the semi-strip's lateral sides, important to engineering applications, are investigated. [ABSTRACT FROM AUTHOR]

In this article we consider moduli properties of singular curves on K3 surfaces. Let B g denote the stack of primitively polarized K3 surfaces ( X , L ) of genus g and let T g , k n → B g be the stack parameterizing tuples [ ( f : C → X , L ) ] with f an unramified morphism which is birational onto its image, C a smooth curve of genus p ( g , k ) − n and f ⁎ C ∈ | k L | . We show that the forgetful morphism η : T g , k n → M p ( g , k ) − n is generically finite on at least one component, for all but finitely many values of p ( g , k ) − n . We further study the Brill–Noether theory of those curves parametrized by the image of η , and find a Wahl-type obstruction for a smooth curve with an unordered marking to have a nodal model on a K3 surface in such a way that the marking is the divisor over the nodes. [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]

If two closed G -manifolds are G -cobordant then characteristic numbers corresponding to the fixed point sets (submanifolds) of subgroups of G and to normal bundles to these sets coincide. We construct two analogues of these characteristic numbers for singular complex G -varieties where G is a finite group. They are defined as sums of certain indices of collections of 1-forms (with values in the spaces of the irreducible representations of subgroups). These indices are generalizations of the GSV-index (for isolated complete intersection singularities) and the Euler obstruction respectively. [ABSTRACT FROM AUTHOR]

An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consisting of a sheaf of ideals and a positive integer. This approach is particularly simple and seems suitable for applications to a good theory of simultaneous algorithmic resolution of singularities, specially for families parametrized by the spectrum of an artinian ring. [ABSTRACT FROM AUTHOR]

This article provides a simple presentation of an algorithm to resolve singularities of algebraic varieties over fields of characteristic zero by means of a sequence of blowing ups with smooth centers contained in the set of points of maximum multiplicity. The algorithm uses primarily multiplicity, rather than the Hilbert–Samuel function, to control the resolution process, and it does not involve a local embedding into a smooth variety. The paper introduces a generalization of the usual notion of equivalence in the theory of resolution of singularities, which is important to justify an essential step in the construction of the algorithm. [ABSTRACT FROM AUTHOR]

Given an ideal a ⊆ R in a (log) Q -Gorenstein F -finite ring of characteristic p > 0 , we study and provide a new perspective on the test ideal τ ( R , a t ) for a real number t > 0 . Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ ( R , a t ) using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F -jumping numbers of τ ( R , a t ) as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals. [ABSTRACT FROM AUTHOR]

In this paper we prove a version of curved Koszul duality for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}$-graded curved coalgebras and their cobar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations $\mathsf{MF}(R,W)$. We show how Dyckerhoff’s generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel–Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category $\mathsf{MF}(R,W)$. Similar results are also obtained in the orbifold case and in the graded case. [ABSTRACT FROM AUTHOR]

The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Lê cycles of Z; and vice versa: The Lê cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding Milnor classes. [ABSTRACT FROM AUTHOR]

We prove an analog of the holomorphic Lefschetz formula for endofunctors of smooth compact dg-categories. We deduce from it a generalization of the Lefschetz formula of Lunts (J Algebra 356:230-256, ) that takes the form of a reciprocity law for a pair of commuting endofunctors. As an application, we prove a version of Lefschetz formula proposed by Frenkel and Ngô (Bull Math Sci 1(1):129-199, ). Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity $${\varvec{w}}$$ . We apply this formula to get some restrictions on the Betti numbers of a $${\mathbb Z}/2$$ -equivariant module over $$k[[x_1,\ldots ,x_n]]/({\varvec{w}})$$ in the case when $${\varvec{w}}(-x)={\varvec{w}}(x)$$ . [ABSTRACT FROM AUTHOR]

Let $$X$$ be a smooth variety over an algebraically closed field of characteristic $$p > 0, Z$$ a smooth divisor, and $$j: U=X {\setminus } Z \rightarrow X$$ the natural inclusion. We introduce in an axiomatic way the notion of a $$V$$ -filtration on unit $$F$$ -crystals and prove such axioms determine a unique filtration. It is shown that if $$\mathcal M $$ is a tame unit $$F$$ -crystal on $$U$$ , then such a $$V$$ -filtration along $$Z$$ exists on $$j_*\mathcal M $$ . The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton-Kisin Riemann-Hilbert correspondence. A few applications to $$\mathbb A ^1$$ and gluing are then discussed. [ABSTRACT FROM AUTHOR]

We describe a natural decomposition of a normal complex surface singularity ( X, 0) into its 'thick' and 'thin' parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts. By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of ( X, 0) in terms of its topology and a finite list of numerical bilipschitz invariants. [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]

Abstract: We study Noetherian local rings whose all formal fibers are of dimension zero. Universal catenarity and going-up property of the canonical map to the completion are considered. We present several characterizations of these rings, including a characterization of Weierstrass preparation type. A characterization of local rings with going up property by a strong form of Lichtenbaum–Hartshorne Theorem is obtained. As an application, we give an upper bound for dimension of formal fibers of a large class of algebras over these rings. [Copyright &y& Elsevier]

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $${\varphi}$$ with an isolated singularity at 0 in an open subset of $${\mathbb{C}^n}$$ . This threshold is defined as the supremum of constants c > 0 such that $${e^{-2c\varphi}}$$ is integrable on a neighborhood of 0. We relate $${c(\varphi)}$$ to the intermediate multiplicity numbers $${e_j(\varphi)}$$ , defined as the Lelong numbers of $${(dd^c\varphi)^j}$$ at 0 (so that in particular $${e_0(\varphi)=1}$$ ). Our main result is that $${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$$ . This inequality is shown to be sharp; it simultaneously improves the classical result $${c(\varphi)\geqslant 1/e_1(\varphi)}$$ due to Skoda, as well as the lower estimate $${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals. [ABSTRACT FROM AUTHOR]

Abstract: The Euler obstruction of a function f can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we establish several Lê–Greuel type formulas for germs and . We give applications when g is a generic linear form and when f and g have isolated singularities. [Copyright &y& Elsevier]

Abstract: In 2009, de Fernex and Hacon [10] proposed a generalization of the notion of the singularities to normal varieties that are not -Gorenstein. Based on their work, we generalize Kleiman’s transversality theorem to subvarieties with log terminal or log canonical singularities. We also show that some classical varieties, such as generic determinantal varieties, for general smooth curves, and certain Schubert varieties in are log terminal in de Fernex and Hacon’s notion, and canonical with some suitable boundary in the classical sense. [Copyright &y& Elsevier]

Abstract: The singular points of the curve of a hyperbolic form associated with a cyclic weighted shift matrix are examined. It is shown that the singular points of such a curve are real nodes. Some results related the numerical ranges of cyclic weighted shift matrices are presented. In particular, the existence of flat portions on the boundary of the numerical range depends on the reducibility of the hyperbolic form. Further, an algebraic method is provided for the decomposition of reducible form which leads to a criterion for the periodicity of the weights. [Copyright &y& Elsevier]

We prove that the construction of motivic nearby cycles, introduced by Jan Denef and François Loeser, is compatible with the formalism of nearby motives, developed by Joseph Ayoub. Let $$k$$ be an arbitrary field of characteristic zero, and let $$X$$ be a smooth quasi-projective $$k$$ -scheme. Precisely, we show that, in the Grothendieck group of constructible étale motives, the image of the nearby motive associated with a flat morphism of $$k$$ -schemes $$f:X\rightarrow \mathbb A ^1_k$$ , in the sense of Ayoub’s theory, can be identified with the image of Denef and Loeser’s motivic nearby cycles associated with $$f$$ . In particular, it provides a realization of the motivic Milnor fiber of $$f$$ in the “non-virtual” motivic world. [ABSTRACT FROM AUTHOR]

Abstract: Let be a morphism between normal complex varieties, where is Kawamata log terminal. Given any differential form , defined on the smooth locus of , we construct a “pull-back form” on . The pull-back map obtained by this construction is -linear, uniquely determined by natural universal properties and exists even in cases where the image of is entirely contained in the singular locus of . One relevant setting covered by the construction is that where is the inclusion (or normalisation) of the singular locus . As an immediate corollary, we show that differential forms defined on the smooth locus of induce forms on every stratum of the singularity stratification. The same result also holds for many Whitney stratifications. [Copyright &y& Elsevier]

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern–Schwartz–MacPherson class. The strengthened version recovers the geometric deletion–restriction formula of Denham et al. for arrangement complements, and generalizes Kouchnirenko’s theorem on the Newton polytope for nondegenerate hypersurfaces. [ABSTRACT FROM AUTHOR]

Abstract: We prove a new formula for the Hirzebruch–Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern–Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusiński and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern–Milnor classes of complete intersections with isolated singularities. [Copyright &y& Elsevier]

We prove that for N≥4, all smooth hypersurfaces of degree N in ℙ are birationally superrigid. First discovered in the case N=4 by Iskovskikh and Manin in a work that started this whole direction of research, this property was later conjectured to hold in general by Pukhlikov. The proof relies on the method of maximal singularities in combination with a formula on restrictions of multiplier ideals. [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]

Abstract: We generalize -signature to pairs where is a Cartier subalgebra on as defined by the first two authors. In particular, we show the existence and positivity of the -signature for any strongly -regular pair. In one application, we answer an open question of Aberbach and Enescu by showing that the -splitting ratio of an arbitrary -pure local ring is strictly positive. Furthermore, we derive effective methods for computing the -signature and the -splitting ratio in the spirit of the work of R. Fedder. [Copyright &y& Elsevier]

Suppose R is a d-dimensional reduced F-finite Noetherian local ring with prime characteristic p>0 and perfect residue field. Let $R^{1/p^{e}}$ be the ring of p-th roots of elements of R for e∈ℕ, and let a denote the maximal rank of a free R-module appearing in a direct sum decomposition of $R^{1/p^{e}}$. We show the existence of the limit $s(R) := \lim_{e \to\infty} \frac{a_{e}}{p^{ed}}$, called the F-signature of R. This invariant-which can be extended to all local rings in prime characteristic-was first formally defined by C. Huneke and G. Leuschke (in Math. Ann. 324(2), 391-404, ) and has previously been shown to exist only in special cases. The proof of our main result is based on the development of certain uniform Hilbert-Kunz estimates of independent interest. Additionally, we analyze the behavior of the F-signature under finite ring extensions and recover explicit formulae for the F-signatures of arbitrary finite quotient singularities. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of a derived splinter. Roughly speaking, a scheme is a derived splinter if it splits off from the coherent cohomology of any proper cover. Over a field of characteristic 0, this condition characterises rational singularities, as suggested by the work of Kovács. Our main theorem asserts that over a field of characteristic p, derived splinters are the same as (underived) splinters, i.e. schemes that split off from any finite cover. Using this result, we answer some questions of Karen Smith concerning the extension of Serre/Kodaira-type vanishing results beyond the class of ample line bundles in positive characteristic; these are purely projective geometric statements independent of singularity considerations. In fact, we can prove ‘up to finite cover’ analogues in characteristic p of many vanishing theorems known in characteristic 0. All these results fit naturally in the study of F-singularities, and are motivated by a desire to understand the direct summand conjecture. [ABSTRACT FROM PUBLISHER]

Abstract: We address Nash problem for surface singularities using wedges. We give a refinement of the characterisation in A. Reguera-López (2006) of the image of the Nash map in terms of wedges. Our improvement consists in a characterisation of the bijectivity of the Nash mapping using wedges defined over the base field, which are convergent if the base field is , and whose generic arc has transverse lifting to the exceptional divisor. This improves the results of M. Lejeune-Jalabert and A. Reguera (2008) for the surface case. In the way to do this we find a reformulation of Nash problem in terms of branched covers of normal surface singularities. As a corollary of this reformulation we prove that the image of the Nash mapping is characterised by the combinatorics of a resolution of the singularity, or, what is the same, by the topology of the abstract link of the singularity in the complex analytic case. Using these results we prove several reductions of the Nash problem, the most notable being that, if Nash problem is true for singularities having rational homology sphere links, then it is true in general. [Copyright &y& Elsevier]

Abstract: We investigate a family of conditional independence models defined by constraints on complete but non hierarchical marginal log–linear parameters. By exploiting results on the mixed parameterization, we show that these models are smooth when a certain Jacobian matrix has spectral radius strictly less than 1. In the simple context when only two marginals are involved, we prove that this condition is always satisfied. In the general case, we describe an efficient numerical test for checking whether the condition is satisfied with high probability. This approach is applied to several examples of non hierarchical conditional independence models and to a directed cyclic graph model; we establish that they are all smooth. [Copyright &y& Elsevier]

Abstract: Let k be a commutative ring and A a commutative k-algebra. Given a positive integer m, or , we say that a k-linear derivation δ of A is m-integrable if it extends up to a Hasse–Schmidt derivation of A over k of length m. This condition is automatically satisfied for any m under one of the following orthogonal hypotheses: (1) k contains the rational numbers and A is arbitrary, since we can take ; (2) k is arbitrary and A is a smooth k-algebra. The set of m-integrable derivations of A over k is an A-module which will be denoted by . In this paper we prove that, if A is a finitely presented k-algebra and m is a positive integer, then a k-linear derivation δ of A is m-integrable if and only if the induced derivation is m-integrable for each prime ideal . In particular, for any locally finitely presented morphism of schemes and any positive integer m, the S-derivations of X which are locally m-integrable form a quasi-coherent submodule such that, for any affine open sets and , with , we have and for each . We also give, for each positive integer m, an algorithm to decide whether all derivations are m-integrable or not. [Copyright &y& Elsevier]