Extended canonical algebras and Fuchsian singularities

The algebraic theory of fuchsian singularities

This article has the following aims: (1) Extend the notion of fuchsian singularities (of first kind) to base fields of arbitrary characteristic. (2) Discuss their relationship to mathematical objects of a different nature. (3) Provide a purely ring-theoretic characterization of fuchsian singularities. (4) Expoloit their singularity categories and their Grothendieck groups.

Nakayama algebras and Fuchsian singularities

This present paper is devoted to the study of a class of Nakayama algebras $N_n(r)$ given by the path algebra of the equioriented quiver $\mathbb{A}_n$ subject to the nilpotency degree $r$ for each sequence of $r$ consecutive arrows. We show that the Nakayama algebras $N_n(r)$ for certain pairs $(n,r)$ can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras $N_n(r)$ of Fuchsian type, that is, derived equivalent to …

A McKay correspondence for the Poincaré series of some finite subgroups of ${\rm SL}_3(\CC)$

A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that w…

叶状结构的奇点与代数堆的良模空间

Singularities of Foliations and Good Moduli Spaces of Algebraic Stacks

本文基于最小模型程序理论，研究叶状结构的奇点与代数堆的良模空间，证明log-canonical奇点的稳定子群为有限扩张的代数torus，从而局部存在良模空间，且canonical奇点的稳定点locus非空。