Graded matrix factorizations of size two and reduction

TL;DR

This paper explores the relationship between graded matrix factorizations of size two and the reduction of singularities, providing an inverse construction and analyzing exceptional collections in the context of singularity theory.

Contribution

It introduces an inverse to Wall's reduction of singularities using graded matrix factorizations and studies exceptional collections in this framework.

Findings

Established a correspondence between graded matrix factorizations and complete intersection singularities.

Provided an inverse to Wall's singularity reduction process.

Analyzed exceptional collections in the derived category of matrix factorizations.

Abstract

We associate a complete intersection singularity to a graded matrix factorization of size two of a polynomial in three variables. We show that we get an inverse to the reduction of singularities considered by C.T.C.Wall. We study this for the full strongly exceptional collections in the triangulated category of graded matrix factorizations constructed by H.Kajiura, K.Saito, and the second author.