Schubert compactifications of matrix pencils

Abstract

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.