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Let X ⊆ G / B be a Schubert variety in a flag manifold and let π : X ~ → X be a Bott–Samelson resolution of X. In this paper, we prove an effective version of the decomposition theorem for the derived pushforward R π ∗ Q X ~ . As a by-product, we obtain recursive procedure to extract Kazhdan–Lusztig polynomials from the polynomials introduced by Deodhar [7], which does not require prior knowledge of a minimal set. We also observe that any family of equivariant resolutions of Schubert varieties allows to define a new basis in the Hecke algebra and we show a way to compute the transition matrix, from the Kazhdan–Lusztig basis to the new one. [ABSTRACT FROM AUTHOR]

Let $ {\mathbb {C}}^{n+1}_o$ denote the germ of $ {\mathbb {C}}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $ {\mathbb {C}}^{n+1}_o$ and $W$ a deformation of $V$ over $ {\mathbb {C}}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we prove that $W$ is a $\mu$ -constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$ , for example, the topological triviality of the family $W$ and the fact that the natural morphism $(\operatorname {W({\mathbb {C}}_{o})}_{m})_{{\rm red}}\rightarrow {\mathbb {C}}_{o}$ is flat, where $\operatorname {W({\mathbb {C}}_{o})}_{m}$ is the relative space of $m$ -jets. On the way to the proof of our main result, we give a complete answer to a question of Arnold on the monotonicity of Newton numbers in the case of convenient Newton polyhedra. [ABSTRACT FROM AUTHOR]

Let X be a singular algebraic variety defined over a perfect field k, with quotient field K(X). Let s≥2 be the highest multiplicity of X and let Fs(X) be the set of points of multiplicity s. If Y⊂Fs(X) is a regular center and X←X1 is the blow up at Y, then the highest multiplicity of X1 is less than or equal to s. A sequence of blow ups at regular centers Yi⊂Fs(Xi), say X←X1←⋯←Xn, is said to be a simplification of the multiplicity if the maximum multiplicity of Xn is strictly lower than that of X, that is, if Fs(Xn) is empty. In characteristic zero there is an algorithm which assigns to each X a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties β:X′→X of generic rank r≥1 (i.e., [K(X′):K(X)]=r). We will see that, when imposing suitable conditions on β, there is a strong link between the strata of maximum multiplicity of X and X′, say Fs(X) and Frs(X′) respectively. In such case, we will say that the morphism is strongly transversal. When β:X′→X is strongly transversal one can obtain information about the simplification of the multiplicity of X from that of X′ and vice versa. Finally, we will see that given a singular variety X and a finite field extension L of K(X) of rank r≥1, one can construct (at least locally, in étale topology) a strongly transversal morphism β:X′→X, where X′ has quotient field L. [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]

Let X be a variety over a perfect field k and let X ∞ be its space of arcs. Given a closed subset Z of X , let X ∞ Z denote the subscheme of X ∞ consisting of all arcs centered at some point of Z . We prove that Local Uniformization implies that X ∞ Z has a finite number of irreducible components for each closed subset Z of X . In particular, Local Uniformization implies that X ∞ Sing X has a finite number of irreducible components. [ABSTRACT FROM AUTHOR]

Let X be a variety over a field k and let X ∞ be its space of arcs. Let P E be the stable point of X ∞ defined by a divisorial valuation ν E on X . Assuming char k = 0 , if X is smooth at the center of P E , we make a study of the graded algebra associated to ν E and define a finite set whose elements generate a localization of the graded algebra modulo étale covering. This provides an explicit description of a minimal system of generators of the local ring O X ∞ , P E . If X is singular, we obtain generators of P E / P E 2 and conclude that embdim O ( X ∞ ) red , P E = embdim O X ∞ , P E ˆ ≤ k ˆ E + 1 where k ˆ E is the Mather discrepancy of X with respect to ν E . This provides algebraic tools for explicit computations of the local rings O X ∞ , P E ˆ . [ABSTRACT FROM AUTHOR]

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]

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]

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

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]