Related papers: Skew graded $(A_\infty)$ hypersurface singularitie…

Categorical resolutions of singularities are a replacement of resolution of singularities within the realm of triangulated categories. They allow the study of the derived category of a singular variety $X$ via a triangulated category that…

Let $(A,\mathfrak{m})$ be a Henselian Cohen-Macaulay local ring and let CM(A) be the category of maximal Cohen-Macaulay $A$-modules. We construct $T \colon CM(A)\times CM(A) \rightarrow mod(A)$, a subfunctor of $Ext^1_A(-, -)$ and use it to…

It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free…

The purpose of this note is to show that a finitely generated graded module $M$ over $S=k[x_1,\ldots,x_n]$, $k$ a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree ${\rm adeg}(M)$ agrees with ${\rm adeg}(F/{\rm…

Hypersurfaces of manifolds of constant nonzero sectional curvature are classificated according their restricted homogeneous holonomy groups.

This survey presents some recent results of G.-M.Greuel and the author on vector bundles over algebraic curves and on Cohen-Macaulay modules over surface singularities. It is mainly devoted to the classification problems, especially to the…

Let k be an algebraically closed uncountable field of characteristic zero. Let R be a complete local hypersurface over k. Denote by CM(R) the category of maximal Cohen-Macaulay R-modules and by D^{sg}(R) the singularity category of R.…

Let $X$ be a nonsingular projective surface over an algebraically closed field with characteristic zero, and $H_-$ and $H_+$ ample line bundles on $X$ separated by only one wall of type $(c_1,c_2)$. Suppose the moduli scheme $M(H_-)$ of…

In this paper the singular hypersurfaces in $\mathbb{C}\mathrm{P}^4$ of degree $d$ with an isolated singularity are studied. If the singularity is of type $A_{2k+1}$, under the condition $d<(k+5)/2$, a classification of such hypersurfaces…

In this article, we realize skew-gentle algebras as skew-tiling algebras associated to admissible partial triangulations of punctured marked surfaces. Based on this, we establish a bijection between tagged permissible curves and certain…

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional path algebra KQ of a quiver Q. Under certain assumptions on the action of G, we show the existence of a certain kind of…

Recently it has been proved that any arithmetically Cohen-Macaulay (ACM) bundle of rank two on a general, smooth hypersurface of degree at least three and dimension at least four is a sum of line bundles. When the dimension of the…

Let $R$ be a Cohen-Macaulay local domain. In this paper we study the cone of Cohen-Macaulay modules inside the Grothendieck group of finitely generated $R$-modules modulo numerical equivalences, introduced in \cite{CK}. We prove a result…

Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, and let $f\in A_2$ be a central regular element of $A$. The quotient algebra $A/(f)$ is usually called a (noncommutative) quadric hypersurface. In this paper, we use the…

Let $S_\alpha = k\langle x_1,\dots,x_n\rangle /(x_i x_j - \alpha_{ij} x_j x_i)$ be a standard graded skew polynomial algebra over an algebraically closed field $k$ of characteristic not equal to $2$. We show the following results. When $n$…

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is…

Tilting theory is one of the central tools in modern representation theory, in particular in the study of Cohen-Macaulay representations. We study Cohen-Macaulay representations of $\mathbb N$-graded Artin-Schelter Gorenstein algebras $A$…

We prove that on a nonsingular quadric hypersurface Q in $P^4_k$ even CI liaison classes of ACM curves are in bijective correspondence with the stable equivalence classes (up to shift in degree) of maximal Cohen-Macaulay graded modules over…

We calculate the Krull--Gabriel dimension of the functor category of the (stable) category of maximal Cohen--Macaulay modules over hypersurfaces of countable Cohen--Macaulay representation type. We show that the Krull--Gabriel dimension is…