# Relative singular locus and Balmer spectrum of matrix factorizations

## TL;DR

This paper classifies certain subcategories of the derived matrix factorization category for Landau-Ginzburg models on schemes and shows the Balmer spectrum corresponds to a newly introduced relative singular locus.

## Contribution

It introduces the concept of the relative singular locus and establishes a homeomorphism with the Balmer spectrum of matrix factorizations, linking geometric and categorical structures.

## Key findings

- Classification of thick subcategories of ${
m DMF}(X,L,W)$
- Homeomorphism between Balmer spectrum and relative singular locus
- New geometric invariant: the relative singular locus ${
m Sing}(X_0/X)$

## Abstract

For a separated Noetherian scheme $X$ with an ample family of line bundles and a non-zero-divisor $W\in\Gamma(X,L)$ of a line bundle $L$ on $X$, we classify certain thick subcategories of the derived matrix factorization category ${\rm DMF}(X,L,W)$ of the Landau-Ginzburg model $(X,L,W)$. Furthermore, by using the classification result and the theory of Balmer's tensor triangular geometry, we show that the spectrum of the tensor triangulated category $({\rm DMF}(X,L,W), \otimes^{\frac{1}{2}})$ is homeomorphic to the relative singular locus ${\rm Sing}(X_0/X)$, introduced in this paper, of the zero scheme $X_0\subset X$ of $W$.