Conic intersections, Maximal Cohen-Macaulay modules and the Four Subspace problem

TL;DR

This paper classifies indecomposable maximal Cohen-Macaulay modules over the coordinate ring of four generic points in the projective plane by linking the problem to the Four Subspace problem and representation theory of a specific quiver, revealing connections to conic pencils and Ulrich modules.

Contribution

It reduces the classification of MCM modules over a specific ring to a well-known representation theory problem, providing explicit descriptions and applications to Ulrich modules and algebra structures.

Findings

Classification of indecomposable MCM modules via Four Subspace problem

Identification of matrix factorizations with conic pencils

Determination of Ulrich modules over the coordinate ring

Abstract

Let $X$ be a set of $4$ generic points in $\mathbb{P}^2$ with homogeneous coordinate ring $R$. We classify indecomposable graded MCM modules over $R$ by reducing the classification to the Four Subspace problem solved by Nazarova and Gel$'$fand-Ponomarev, or equivalently to the representation theory of the $\widetilde{D}_4$ quiver. In particular, the $\mathbb{P}^1$ tubular family of regular representations corresponds to matrix factorizations of the pencil of conics going through $X$, with smooth conics $Q_{t}$ corresponding to rank one tubes and the singular conics $Q_0, Q_1, Q_{\infty}$ giving the remaining rank two tubes. As applications we determine the Ulrich modules over $R$ and we identify the preprojective algebra of type $\widetilde{D}_4$ as the diagonal part of the Yoneda algebra of a Koszul $R$-module.