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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]

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]

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]

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 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]

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]

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]

Let [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 paper, the classification of hypersurfaces in with an isolated singularity are studied. If the singularity is of type Ak, under certain restrictions of the degree of the hypersurfaces, a classification up to homeomorphism, which is a diffeomorphism on the nonsingular part, is obtained. Examples of cubic hypersurfaces with an A5-singularity are constructed. [ABSTRACT FROM PUBLISHER]