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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]

In this article, we devise a version of topological |$L^2$| -Serre duality for singular complex spaces with arbitrary singularities. This duality is used to deduce various new |$L^2$| -vanishing theorems for the |${\overline{\partial}}$| -equation on singular spaces. It is shown that complex spaces with rational singularities behave quite tame with respect to the |${\overline{\partial}}$| -equation in the |$L^2$| -sense. More precisely: a singular point is rational if and only if the |$L^2$| - |${\overline{\partial}}_s$| -complex is exact in this point. So, we obtain an |$L^2$| - |${\overline{\partial}}$| -resolution of the structure sheaf in rational singular points. [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]

We clarify the relation between the subcategory D ( A) of homological finite objects in D ( A) and the subcategory K ( P) of perfect complexes in D ( A), by giving two classes of abelian categories A with enough projective objects such that D ( A) = K ( P), and finding an example such that D ( A) ≠ K ( P). We realize the bounded derived category D ( A) as a Verdier quotient of the relative derived category D ( A), where C is an arbitrary resolving contravariantly finite subcategory of A: Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension. We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dim T < ∞ such that ⊥ T is finite, then D (mod A) admits a categorical resolution; and that for a CM (Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant. [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]

We present a procedure for computing the log-canonical threshold of an arbitrary ideal generated by binomials and monomials. The computation of the log canonical threshold is reduced to the problem of computing the minimum of a function, which is defined in terms of the generators of the ideal. The minimum of this function is attained at some ray of a fan which only depends on the exponents of the monomials appearing in the generators of the ideal. [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 Nash multiplicity sequence was defined by Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. Hickel generalized this notion and described a sequence of blow ups which allows us to compute it and study its behavior. In this paper, we show how this sequence can be used to compute some invariants that appear in algorithmic resolution of singularities. Moreover, this indicates that these invariants from constructive resolution are intrinsic to the variety since they can be read in terms of its space of arcs. This result is a first step connecting explicitly arc spaces and algorithmic resolution of singularities. [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]

Let G ⊆ GL ( n ) be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution X → C n / G , which is based just on the geometry of the singularity C n / G , without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular . As an application, we determine the Cox rings of resolutions X → C 3 / G for all G ⊆ GL ( 3 ) with the aforementioned property and of order | G | ≤ 12 . We also provide examples in dimension 4. [ABSTRACT FROM AUTHOR]

A 3-fold terminal quotient singularity X = C 3 / G admits the economic resolution Y → X , which is “close to being crepant”. This paper proves that the economic resolution Y is isomorphic to a distinguished component of a fine moduli space of θ -stable G -constellations for a suitable GIT parameter θ . [ABSTRACT FROM AUTHOR]

We develop an equivariant version of the Hirzebruch class for singular varieties. When the group acting is a torus we apply localization theorem of Atiyah-Bott and Berline-Vergne. The localized Hirzebruch class is an invariant of a singularity germ. The singularities of toric varieties and Schubert varieties are of special interest. We prove certain positivity results for simplicial toric varieties. The positivity for Schubert varieties is illustrated by many examples, but it remains mysterious. [ABSTRACT FROM AUTHOR]

We study birational maps among (1) the moduli space of semistable sheaves of Hilbert polynomial $$4m+2$$ on a smooth quadric surface, (2) the moduli space of semistable sheaves of Hilbert polynomial $$m^{2}+3m+2$$ on $$\mathbb {P}^{3}$$ , (3) Kontsevich's moduli space of genus-zero stable maps of degree 2 to the Grassmannian Gr(2, 4). A regular birational morphism from (1) to (2) is described in terms of Fourier-Mukai transforms. The map from (3) to (2) is Kirwan's partial desingularization. We also investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs. [ABSTRACT FROM AUTHOR]

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety. [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]

Let X be an equidimensional scheme of finite type over a perfect field k . Under these conditions, the multiplicity along points of X defines an upper semi-continuous function, say mult X : X → N , which stratifies X into its locally closed level sets. We study this stratification, and the behavior of the multiplicity when blowing up at regular equimultiple centers. We also discuss a natural compatibility of these two concepts when X is replaced with its underlying reduced scheme. The main result in this paper is to show that, given a variety X , there is a well defined Rees algebra over X , naturally attached to maximum value of the multiplicity. [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]

In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but generalizes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring R, that is, the set of all possible finite global dimensions of endomorphism rings of MCM R-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings. [ABSTRACT FROM AUTHOR]

In his previous article, the author has defined a higher version of the Nash blowup. In this article, we will introduce another higher version and prove that it is compatible with products and smooth morphisms. We will also prove that the product of curves can be desingularized via both versions. [ABSTRACT FROM AUTHOR]