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Let (V, 0) be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : (C n , 0) → (C , 0) . We introduce a new derivation Lie algebra associated to (V, 0). The new Lie algebra is defined by the ideal of antiderivatives with respect to the Tjurina ideal of (V, 0). More precisely, let I = (f , ∂ f ∂ x 1 , … , ∂ f ∂ x n ) and Δ (I) : = { g ∣ g , ∂ g ∂ x 1 , … , ∂ g ∂ x n ∈ I } , then A Δ (V) : = O n / Δ (I) and L Δ (V) : = Der (A Δ (V) , A Δ (V)) . Their dimensions as a complex vector space are denoted as β (V) and δ (V) , respectively. δ (V) is a new invariant of singularities. In this paper we study the new local algebra A Δ (V) and the derivation Lie algebra L Δ (V) , and also compute them for fewnomial isolated singularities. Moreover, we formulate sharp lower estimation conjectures for β (V) and δ (V) when (V, 0) are weighted homogeneous isolated hypersurface singularities. We verify these conjectures for a large class of singularities. Lastly, we provide an application of β (V) and δ (V) to distinguishing contact classes of singularities. [ABSTRACT FROM AUTHOR]

Following Shokurov's idea, we give a simple proof of the ACC conjecture for minimal log discrepancies for surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new invariant to isolated complete intersection singularities. We use this new invariant to obtain two characterization theorems for contact simple complete intersection singularities. [ABSTRACT FROM AUTHOR]

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps. [ABSTRACT FROM AUTHOR]

For i : Z ↪ X a closed immersion of smooth varieties, we study how the V-filtration along Z and the Hodge filtration on a mixed Hodge module M on X interact with each other. We also give a formula for the functors i ∗ , i ! in terms of this V-filtration. As applications, we obtain results on the Hodge filtration of monodromic mixed Hodge modules and we give a Hodge theoretic proof of Skoda's theorem on multiplier ideals. Finally, we use the results to study the Fourier–Laplace transform of a monodromic mixed Hodge module. [ABSTRACT FROM AUTHOR]

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in C n which is Lipschitz regular at infinity must be an affine linear subspace of C n . In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if X and Y are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then X is a complex algebraic set if and only if Y is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Fukui–Kurdyka–Paunescu conjecture, which says that sub-analytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (respectively, degree) of real and complex analytic (respectively, algebraic) sets. For instance, still in the real case, we prove a global version of the Fukui–Kurdyka–Paunescu conjecture. In the complex case, one of the results that we prove is the following: if $(X,0)\subset (\mathbb {C}^{n},0)$ , $(Y,0)\subset (\mathbb {C}^{m},0)$ are germs of analytic sets and $h\colon (X,0)\to (Y,0)$ is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs $(X,0)$ and $(Y,0)$ have the same multiplicity. One of the results that we prove in the global case is the following: if $X\subset \mathbb {C}^{n}$ , $Y\subset \mathbb {C}^{m}$ are algebraic sets and $\phi \colon X\to Y$ is a semi-algebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in $\mathbb {P}^{n+m}(\mathbb {C})$ is an orientable homological cycle, then ${\rm deg}(X)={\rm deg}(Y)$. [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]

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 study the motivic and ℓ -adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D. Orlov and J. Burke–M. Walker to give a formula for the ℓ -adic realization of the dg category of singularities of the zero locus of a global section of a vector bundle. In particular, we obtain a formula for the ℓ -adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension n. [ABSTRACT FROM AUTHOR]

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities. [ABSTRACT FROM AUTHOR]

We undertake a systematic study of Lipschitz normally embedded normal complex surface germs. We prove, in particular, that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz normally embedded surface singularity. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Kollár stating that the local fundamental group of a klt singularity x is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of x is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair. [ABSTRACT FROM AUTHOR]

Given a logarithmic 1‐form on the snc locus of a log canonical surface pair (X,D) over a perfect field of characteristic p≥7, we show that it extends with at worst logarithmic poles to any resolution of singularities. We also prove the analogous statement for regular differential forms, under an additional tameness hypothesis. In addition, residue and restriction sequences for tamely dlt pairs are established. We give a number of examples showing that our results are sharp in the surface case, and that they fail in higher dimensions. On the other hand, our techniques yield a new proof of the characteristic zero Logarithmic Extension Theorem in any dimension. [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]

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$ -finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama's description of the restricted base locus to klt or strongly $F$ -regular varieties over arbitrary fields. [ABSTRACT FROM AUTHOR]

In this note we give a simple proof of the following relative analog of the well known Milnor-Palamodov theorem: the Bruce-Roberts number of a function relative to an isolated hypersurface singularity is equal to its topological Milnor number (the rank of a certain relative (co)homology group) if and only if the hypersurface singularity is quasihomogeneous. The proof relies on an interpretation of the Bruce-Roberts number in terms of differential forms and the Lê-Greuel formula. [ABSTRACT FROM AUTHOR]

Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematics we are interested in is implemented in the computer algebra system Singular, whose focus is on polynomial computations, while the coordination is left to the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of Singular, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka's celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework and investigate how the computations scale up to 256 cores. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero, X and Y be smooth K‐varieties, and let G be an algebraic K‐group. Given two algebraic morphisms φ:X→G and ψ:Y→G, we define their convolution φ∗ψ:X×Y→G by φ∗ψ(x,y)=φ(x)·ψ(y). We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism φ:X→G which is dominant when restricted to each geometrically irreducible component of X, by convolving it with itself finitely many times, one obtains a flat morphism with reduced fibers of rational singularities. Uniform bounds on families of morphisms are given as well. Moreover, as a key analytic step, we also prove the following result in motivic integration; if {fQp:Qpn→C}p∈primes is a collection of motivic functions, and fQp is L1 for any p large enough, then in fact there exists ε>0 such that fQp is L1+ε for any p large enough. [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]

The purpose of this note is to correct a mistake in the article "A curve selection lemma in spaces of arcs and the image of the Nash map" Compositio Math. 142 (2006), 119–130. It is due to an overlooked hypothesis in the definition of generically stable subset of the space of arcs X∞ of a variety X defined over a perfect field k. [ABSTRACT FROM AUTHOR]

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$. The Yau algebra $L(V)$ is defined to be the Lie algebra of derivations of the moduli algebra $A(V):={\mathcal{O}}_{n}/(f,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{1},\ldots ,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{n})$ , that is, $L(V)=\text{Der}(A(V),A(V))$. It is known that $L(V)$ is finite dimensional and its dimension $\unicode[STIX]{x1D706}(V)$ is called the Yau number. We introduce a new series of Lie algebras, that is, $k$ th Yau algebras $L^{k}(V)$ , $k\geq 0$ , which are a generalization of the Yau algebra. The algebra $L^{k}(V)$ is defined to be the Lie algebra of derivations of the $k$ th moduli algebra $A^{k}(V):={\mathcal{O}}_{n}/(f,m^{k}J(f)),k\geq 0$ , that is, $L^{k}(V)=\text{Der}(A^{k}(V),A^{k}(V))$ , where $m$ is the maximal ideal of ${\mathcal{O}}_{n}$. The $k$ th Yau number is the dimension of $L^{k}(V)$ , which we denote by $\unicode[STIX]{x1D706}^{k}(V)$. In particular, $L^{0}(V)$ is exactly the Yau algebra, that is, $L^{0}(V)=L(V),\unicode[STIX]{x1D706}^{0}(V)=\unicode[STIX]{x1D706}(V)$. These numbers $\unicode[STIX]{x1D706}^{k}(V)$ are new numerical analytic invariants of singularities. In this paper we formulate a conjecture that $\unicode[STIX]{x1D706}^{(k+1)}(V)>\unicode[STIX]{x1D706}^{k}(V),k\geq 0.$ We prove this conjecture for a large class of singularities. [ABSTRACT FROM AUTHOR]

We present a probabilistic Las Vegas algorithm for solving sufficiently generic square polynomial systems over finite fields. We achieve a nearly quadratic running time in the number of solutions, for densely represented input polynomials. We also prove a nearly linear bit complexity bound for polynomial systems with rational coefficients. Our results are obtained using the combination of the Kronecker solver and a new improved algorithm for fast multivariate modular composition. [ABSTRACT FROM AUTHOR]

Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions. [ABSTRACT FROM AUTHOR]

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications. [ABSTRACT FROM AUTHOR]

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities. A concrete application is for surfaces with geometric genus pg=5: the canonical model is embedded in P4 if and only if we have a complete intersection of type (2,4) or (3,3). [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]

For an analytic set V in K n , K = R or C , which contains the origin 0 ∈ K n , the geometric tangent cone of V at 0 is the set of vectors in K n which are the limits of secant lines passing through the origin and non-zero sequences in V that converge to the origin; the algebraic tangent cone of V at 0 is the algebraic set defined by the ideal generated by the initial forms of all analytic functions f whose germs at 0 are in the ring of germs of analytic functions in K n about 0 which vanish on the germ of V at 0. In this paper, we give some characterization for the geometric tangent cone and compare these two tangent cones of V at 0. In particular, we characterize the geometric tangent cone of V at 0 via the so-called Ł ojasiewicz exponent of an analytic map germ along a line and compute this number in some special cases. [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]

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depth R / I (t) = dim R - pd I (t) - 1 , where I (t) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and pd denotes the projective dimension. It has been an open question whether the function depth R / I (t) is non-increasing if I is a squarefree monomial ideal. We show that depth R / I (t) is almost non-increasing in the sense that depth R / I (s) ≥ depth R / I (t) for all s ≥ 1 and t ∈ E (s) , where E (s) = ⋃ i ≥ 1 { t ∈ N | i (s - 1) + 1 ≤ t ≤ i s } (which contains all integers t ≥ (s - 1) 2 + 1 ). The range E(s) is the best possible since we can find squarefree monomial ideals I such that depth R / I (s) < depth R / I (t) for t ∉ E (s) , which gives a negative answer to the above question. Another open question asks whether the function depth R / I (t) is always constant for t ≫ 0 . We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that I (t) is integrally closed for t ≫ 0 (e.g. if I is a squarefree monomial ideal), then depth R / I (t) is constant for t ≫ 0 with lim t → ∞ depth R / I (t) = dim R - dim ⊕ t ≥ 0 I (t) / m I (t). Our last result (which is the main contribution of this paper) shows that for any positive numerical function ϕ (t) which is periodic for t ≫ 0 , there exist a polynomial ring R and a homogeneous ideal I such that depth R / I (t) = ϕ (t) for all t ≥ 1 . As a consequence, for any non-negative numerical function ψ (t) which is periodic for t ≫ 0 , there is a homogeneous ideal I and a number c such that pd I (t) = ψ (t) + c for all t ≥ 1 . [ABSTRACT FROM AUTHOR]

We study real analytic isolated singularities of type f:(Cn+m+1,0)→(C,0) with f(z,w)=g(z)+∑i=1m+1hi(wi,w¯i), where g is holomorphic and each hi is a mixed polynomial, with z=(z1,⋯,zn) and w=(w1,⋯,wm+1). We construct a Lê's vanishing polyhedron for f, which describes the degeneration of its Milnor fiber Ff to the singular fiber. Then we prove that Ff is homotopy equivalent to the join Fg∗Fh1∗⋯∗Fhm+1, where Fg is the Milnor fiber of g and Fhi is the Milnor fiber of hi. This implies that Ff has the homotopy type of a bouquet of spheres Sn+m. So we can define the Milnor number μ(f) as the number of spheres in that bouquet, as in the complex setting. [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]

In this paper, we characterize the representation type of an acyclic quiver by the properties of its associated quiver Grassmannians. This characterization utilizes and extends known results about singular quiver Grassmannians and cell decompositions into affine spaces. While all quiver Grassmannians for indecomposable representations of quivers of finite representation type are smooth and admit cell decompositions, it turns out that all quiver Grassmannians for indecomposable representations of tame quivers admit cell decompositions, but some of these quiver Grassmannians are singular (even as varieties). A quiver is wild if and only if there exists a quiver Grassmannian with negative Euler characteristic. [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 prove that a strongly F-regular scheme X admits a finite, generically Galois, and étale-in-codimension-one cover ~X → X such that the étale fundamental groups of ~X and ~~Xreg agree. Equivalently, every finite étale cover of ~Xreg extends to a finite étale cover of ~X. This is analogous to a result for complex klt varieties by Greb, Kebekus, and Peternell. [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]

The Milnor number of an isolated hypersurface singularity, defined as the codimension μ(f) of the ideal generated by the partial derivatives of a power series f that represents locally the hypersurface, is an important topological invariant of the singularity over the complex numbers. However it may loose its significance when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number depends upon the equation f representing the hypersurface, hence it is not an invariant of the hypersurface. For a plane branch represented by an irreducible convergent power series f in two indeterminates over the complex numbers, it was shown by Milnor that μ(f) always coincides with the conductor c(f) of the semigroup of values S(f) of the branch. This is not true anymore if the characteristic of the ground field is positive. In this paper we show that, over algebraically closed fields of arbitrary characteristic, this is true, provided that the semigroup S(f) is tame, that is, the characteristic of the field does not divide any of its minimal generators. [ABSTRACT FROM AUTHOR]

In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin–Schelter regular algebras A of global dimension two, where A is a graded H -module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring A H under such an action arises as an analogue of a coordinate ring of a Kleinian singularity. [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic zero. Let V be an integral affine k -variety. We prove that any torsion Kähler differential form ω ∈ Ω V / k 1 ( V ) vanishes at every formal germ of curve on V . If V is an integral hypersurface of A k N ( N ≥ 1 ), we also prove that the non-cylindrical singular points of V belong to the singular locus of any torsion Kähler differential forms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the question of when the associated graded ring along a valuation, gr ν ∗ ( S ), is a finite gr ν ∗ ( R )‐module, where S is a normal local ring which lies over a normal local ring R and ν ∗ is a valuation of the quotient field of S which dominates S. We begin by discussing some examples and results, allowing us to refine the conditions under which finite generation can hold. We must impose the condition that the extension of valuations is defectless and performs a birational extension of R along the valuation to obtain finite generation (replacing S with the local ring of the quotient field of S determined by the valuation which lies over the extension of R). With these assumptions, we have that finite generation holds, when R is a two‐dimensional excellent local ring. Our main result (in Theorem  ) is to show that for an arbitrary valuation in an algebraic function field over an arbitrary field of characteristic zero, after a birational extension along the valuation, we always have finite generation (all finite extensions of valued fields are defectless in characteristic zero). This generalizes an earlier result, in [Cutkosky, Math. Proc. Cambridge Soc. 106 (2016) 233–255], showing that finite generation holds (after a birational extension) with the additional assumptions that ν has rank 1 and has an algebraically closed residue field. There are essential difficulties in removing these assumptions, which are addressed in the proof in this paper. We obtain general results for unramified extensions of excellent local rings in Proposition  , showing that after blowing up, the extension of associated graded rings is finitely generated of an extremely simple form. This proposition plays an essential role in the Proof of Theorem  . [ABSTRACT FROM AUTHOR]

For an affine toric variety Spec ( A ) , we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Under certain assumptions we compute the dimensions of the Hodge summands T ( i ) 1 ( A ) , generalizing the existing results about the André–Quillen cohomology group T ( 1 ) 1 ( A ) . We prove that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization. [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]

In this note, we show that the free generators of the Mishchenko-Fomenko subalgebra of a complex reductive Lie algebra, constructed by the argument shift method at a regular element, form a regular sequence. This result was proven by Serge Ovsienko in the type A at a regular and semisimple element. Our approach is very different, and is strongly based on geometric properties of the nilpotent bicone. [ABSTRACT FROM AUTHOR]

Boix, De Stefani and Vanzo have characterised ordinary/supersingular elliptic curves over F p in terms of the level of the defining cubic homogeneous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve C of genus g ≥ 2 has level 2. We provide a good number of examples and raise a conjecture. [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 Λ be a concealed canonical algebra and d the dimension vector of a Λ-module which is periodic respect to the action of the Auslander–Reiten translation τ . In the paper, we investigate the union of the closures of the orbits of the τ -periodic Λ-modules of dimension vector d . We show that this set is closed and regular in codimension one. [ABSTRACT FROM AUTHOR]

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]