Pencils of singular quadrics of constant rank and their orbits

TL;DR

This paper provides a geometric classification of singular pencils of quadrics with constant rank and analyzes their orbit structure within the Grassmannian under group actions.

Contribution

It introduces a geometric framework linking singular quadrics of constant rank to vector bundle splitting types and studies their orbit structure in the Grassmannian.

Findings

Characterization of singular pencils via vector bundle splitting

Description of orbit structure in the Grassmannian

Connection between geometric properties and group actions

Abstract

We give a geometric description of singular pencils of quadrics of constant rank, relating them to the splitting type of some naturally associated vector bundles on $\mathbb{P}^1$. Then we study their orbits in the Grassmannian of lines, under the natural action of the general linear group.