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In this paper we explore the generalized Nash problem for arcs on a germ of smooth surface: given two prime divisors above its special point, to determine whether the arc space of one of them is included in the arc space of the other one. We prove that this problem is combinatorial and we explore its relation with several notions of adjacency of plane curve singularities. [ABSTRACT FROM AUTHOR]

In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure B ( 1 ) ‾ of type A, and study relations between an NCCR and crepant resolutions Y and Y + of B ( 1 ) ‾ . More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions Y and Y + of B ( 1 ) ‾ as moduli spaces of representations of the quiver. We also study the Kawamata–Namikawa's derived equivalence between crepant resolutions Y and Y + of B ( 1 ) ‾ in terms of an NCCR. We also show that the P-twist on the derived category of Y corresponds to a certain operation of the NCCR, which we call multi-mutation , and that a multi-mutation is a composition of Iyama–Wemyss's mutations. [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]

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]

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: 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]

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]

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]

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: 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]

Abstract: The Milnor–Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincaré dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet–Schürmann–Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor–Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of vanishing cycles and singular strata in a Whitney stratification of X. Our approach is based on Schürmann''s specialization property for the motivic Hirzebruch class transformation of Brasselet–Schürmann–Yokura. The present results also yield calculations of Todd, Chern and L-type characteristic classes of hypersurfaces. [Copyright &y& Elsevier]

Abstract: We prove that every globally F-regular variety is log Fano. In other words, if a prime characteristic variety X is globally F-regular, then it admits an effective -divisor Δ such that is ample and has controlled (Kawamata log terminal, in fact globally F-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon''s new point of view on Kawamata log terminal singularities in the non--Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally F-regular type. Our techniques apply also to F-split varieties, which we show to satisfy a “log Calabi–Yau” condition. We also prove a Kawamata–Viehweg vanishing theorem for globally F-regular pairs. [Copyright &y& Elsevier]

Abstract: Let V be a finite set of divisorial valuations centered at a 2-dimensional regular local ring R. In this paper we study its structure by means of the semigroup of values, , and the multi-index graded algebra defined by V, . We prove that is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in V, the approximation of a reduced plane curve singularity C by families of sets of divisorial valuations, and the relationship between the value semigroup of C and the semigroups of the sets , allow us to obtain the (finite) minimal generating sequences for C as well as for V. We also analyze the structure of the homogeneous components of . The study of their dimensions allows us to relate the Poincaré series for V and for a general curve C of V. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A''Campo type for the Poincaré series of V. Moreover, the Poincaré series of C could be seen as the limit of the series of , . [Copyright &y& Elsevier]

Abstract: Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive R-module is free. [Copyright &y& Elsevier]

Abstract: Let X denote an equivariant embedding of a connected reductive group G over an algebraically closed field k. Let B denote a Borel subgroup of G and let Z denote a -orbit closure in X. When the characteristic of k is positive and X is projective we prove that Z is globally F-regular. As a consequence, Z is normal and Cohen–Macaulay for arbitrary X and arbitrary characteristics. Moreover, in characteristic zero it follows that Z has rational singularities. This extends earlier results by the second author and M. Brion. [Copyright &y& Elsevier]

Abstract: Various topological properties of projective duality between real projective varieties and their duals are obtained by making use of the microlocal theory of (subanalytically) constructible sheaves developed by Kashiwara [M. Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985) 193–209] and Kashiwara–Schapira [M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss., vol. 292, Springer, Berlin–Heidelberg–New York, 1990]. In particular, we prove in the real setting some results similar to the ones proved by Ernström in the complex case [L. Ernström, Topological Radon transforms and the local Euler obstruction, Duke Math. J. 76 (1994) 1–21]. For this purpose, we describe the characteristic cycles of topological Radon transforms of constructible functions in terms of curvatures of strata in real projective spaces. [Copyright &y& Elsevier]