Schubert compactifications of matrix pencils
Abstract
The Schubert compactification of a linear space of matrices L ⊆ Matn× m is a natural compactification of L in the Grassmannian 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 ℙ2. 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 GL2 × GLn ×GLm. 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 Np in a certain range.
Related
- Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties
- Genus-3 Lefschetz pencils on torus bundles
- Reducibility and Diagonal Equality Strata for Finite Jacobi Pencils
- Dilation-Covariant Hankel Pencils for Multiscale Recovery of Sparse Mellin Spectra
- Adjoint Closures of Singular Quadratic Pencils and First's Pfister-Type Conjecture
BibTeX
@misc{galgano2026schubert,
title = {{Schubert compactifications of matrix pencils}},
author = {Vincenzo Galgano and Fulvio Gesmundo and Hanieh Keneshlou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07164},
eprint = {2610.07164},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07164},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Schubert compactifications of matrix pencils
AU - Vincenzo Galgano
AU - Fulvio Gesmundo
AU - Hanieh Keneshlou
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07164
UR - https://arxiv.org/abs/2610.07164
AB - The Schubert compactification of a linear space of matrices L ⊆ Mat_n× m is a natural compactification of L in the Grassmannian 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 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 GL_2 × GL_n ×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.
ER -