Triangle singularities, ADE-chains, and weighted projective lines

TL;DR

This paper explores the connection between triangle singularities, ADE-chains, and weighted projective lines, revealing their categorical structures, Calabi-Yau properties, and symmetries related to Nakayama algebras.

Contribution

It establishes a link between the stable categories of vector bundles on weighted projective lines and singularity categories, introducing new categorical equivalences and symmetries.

Findings

The stable category is fractional Calabi-Yau with a dimension depending on the Euler characteristic.

Existence of a tilting object shaped as an $(a-1)(b-1)(c-1)$-cuboid.

Identification of ADE-chains extending classical Dynkin cases to new triangulated categories.

Abstract

We investigate the triangle singularity $f=x^a+y^b+z^c$, or $S=k[x,y,z]/(f)$, attached to a weighted projective line $X$ given by the weight triple $(a,b,c)$. We investigate the stable category of vector bundles on $X$ obtained from the vector bundles by factoring out all line bundles. This category is triangulated and has Serre duality. It is, moreover, naturally equivalent to the stable category of graded maximal Cohen-Macaulay modules over $S$ (or matrix factorizations of $f$), and then by results of Buchweitz and Orlov to the graded singularity category of $f$. We show that this category is fractional Calabi-Yau with a CY-dimension that is a function of the Euler characteristic of $X$. We show the existence of a tilting object which has the shape of an $(a-1)(b-1)(c-1)$-cuboid. Particular attention is given to the weight types $(2,a,b)$, yielding an explanation of Happel-Seidel symmetry for a class of important Nakayama algebras. In particular, the weight sequence $(2,3,p)$ corresponds to an ADE-chain, the $E_n$-chain, extrapolating the exceptional Dynkin cases $E_6$, $E_7$ and $E_8$ to a whole sequence of triangulated categories.