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

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.