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arXiv 1702.06608math.ACmath.RT
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圆锥交、极大Cohen-Macaulay模与四子空间问题

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

Vincent Gelinas

备注：
23 pages. Fixed minor issues and added Ulrich modules. Comments welcome
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摘要：

令$X$是具有齐次坐标环$R$的$\mathbb{P}^2$中的$4$类属点的集合。我们分类不可分解的分次MCM模块在$R$通过减少的分类Nazarova和凝胶$'$ fand-Ponomarev解决的四子空间问题，或等价的表示理论的$\widetilde{D}_4$的。特别地，正则表示的$\mathbb{P}^1$管状族对应于通过$X$的二次曲线束的矩阵因式分解，其中光滑二次曲线$Q_{t}$对应于秩1管，奇异二次曲线$Q_0, Q_1, Q_{\infty}$给出剩余秩2管。作为应用，我们确定了$R$上的Ulrich模，并证明了$\widetilde{D}_4$型预投射代数是Koszul $R$ -模的Yoneda代数的对角部分.

英文摘要：

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.

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