Matrix factorizations for domestic triangle singularities

TL;DR

This paper explicitly determines matrix factorizations for domestic triangle singularities over algebraically closed fields, linking algebraic, geometric, and representation-theoretic methods to classify these singularities.

Contribution

It provides a new, explicit description of matrix factorizations for domestic triangle singularities using projective covers and representation theory, differing from prior approaches.

Findings

Explicit matrix factorizations with scalar monomials

Connection to weighted projective lines and Dynkin diagrams

Symmetric factorizations with scalars in {0, ±1}

Abstract

Working over an algebraically closed field $k$ of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities $f=x^a+y^b+z^c$ of domestic type, that is, we assume that $(a,b,c)$ are integers at least two, satisfying $1/a+1/b+1/c>1$. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type $(a,b,c)$. Equivalently, in a representation-theoretic context, we can work in the mesh category of $\mathbb{Z}\tilde\Delta$ over $k$, where $\tilde\Delta$ is the extended Dynkin diagram, corresponding to the Dynkin diagram $\Delta=[a,b,c]$. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the $\mathbb{Z}$-graded simple singularities by Kajiura-Saito-Takahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar multiples of monomials, with scalars taken from $\{0,\pm1\}$.