A category of graded matrix factorizations of a deformed polynomial associated to the $A_{\mu}$-singularity

TL;DR

This paper constructs a full strongly exceptional collection in the triangulated category of graded matrix factorizations for a deformed polynomial related to the $A_{9}$-singularity, using a formal variable to ensure homogeneity.

Contribution

It introduces a method to build a full strongly exceptional collection in the category of graded matrix factorizations for deformed $A_{9}$-singularities with a fixed parameter.

Findings

Constructed a full strongly exceptional collection for generic parameters.

Established a framework for graded matrix factorizations of deformed polynomials.

Utilized a formal variable to homogenize the polynomial.

Abstract

We discuss a triangulated category of graded matrix factorizations of a deformed polynomial associated to the $A_{\mu}\textrm{-}$singularity. The semi-universal deformation of the $A_{\mu}\textrm{-}$singularity is given by a certain deformation of the polynomial of type $A_{\mu}$. In this paper, we consider the category of graded matrix factorizations associated to this deformed polynomial for a fixed parameter. To do so, we introduce a formal variable to make the polynomial homogeneous. As our main result, we construct a full strongly exceptional collection in this category for a generic parameter.