发表机构
Max Planck Institute of Molecular Cell Biology and Genetics; Technische Universität Dresden; Université de Toulouse; CNRS; Universität Würzburg（马克斯·普朗克分子细胞生物学与遗传学研究所; 德累斯顿工业大学; 图卢兹大学; 法国国家科学研究中心; 维尔茨堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对矩阵束（维数为2的线性矩阵空间）构造其Schubert紧化，得到具有商奇点的正规曲面，并利用Kronecker不变量计算其度、上同调环及有理曲线相交数，证明曲面可决定束的正则分量，且其Plücker理想由二次生成并满足Green性质。
AI 中文摘要
矩阵空间 $L \subseteq \textrm{Mat}_{n\times m}$ 的Schubert紧化是 $L$ 在Grassmannian $\textrm{Gr}(n, n+m)$ 中的一个自然紧化。该构造推广了经典的Schubert簇和拟阵Schubert簇，并将线性矩阵空间的不变量转化为相关簇的射影与相交理论不变量。我们在矩阵束的情形（即 $\dim L=2$）下完整地发展了这一几何理论。所得的Schubert曲面是作为 $\mathbb{P}^2$ 的爆破得到的 rational 曲面的奇异类比。我们证明了它们是具有商奇点的正规曲面，计算了它们的次数、上同调环，并描述了特殊的 rational 曲线及其相交配对，这些都用矩阵束的Kronecker不变量来表达。值得注意的是，这一字典可以反向使用：束的不变量可以从簇的相交数和其他不变量中读出。特别地，我们证明了Schubert曲面在 $\mathrm{GL}_2 \times \mathrm{GL}_n \times\mathrm{GL}_m$ 的自然作用下决定了束的正则分量。最后，我们证明了矩阵束的Schubert曲面在Plücker嵌入中的理想由次数 $2$ 的元素生成，并在一定范围内满足Green性质 $N_p$。
英文摘要
The Schubert compactification of a linear space of matrices $L \subseteq \textrm{Mat}_{n\times m}$ is a natural compactification of $L$ in the Grassmannian $\textrm{Gr}(n, n+m)$. This construction generalises classical Schubert varieties and matroid Schubert varieties, and it turns invariants of linear matrix spaces into projective and intersection-theoretic invariants of the associated variety. We develop this geometry completely in the case of matrix pencils, that is $\dim L=2$. The resulting Schubert surfaces are singular analogues of rational surfaces obtained as blow-ups of $\mathbb{P}^2$. We show that they are normal with quotient singularities, compute their degree, cohomology ring, and describe the distinguished rational curves and their intersection pairing in terms of the Kronecker invariants of the matrix pencil. Remarkably, this dictionary can be reversed: the invariants of the pencil are read off from intersection numbers and other invariants of the variety. In particular, we prove that the Schubert surface determines the regular component of the pencil, up to the natural action of $\mathrm{GL}_2 \times \mathrm{GL}_n \times\mathrm{GL}_m$. Finally, we prove that the ideal of Schubert surfaces of matrix pencils in the Plücker embedding is generated in degree $2$ and satisfies Green's property $N_p$ in a certain range.
Comments45 pages, 4 figures. Comments are welcome!
