On invariant rank two vector bundles on $\mathbb{P}^2$

TL;DR

This paper characterizes rank two vector bundles on the projective plane that are invariant under certain subgroup actions of PGL(3), revealing that the geometric jumping locus does not fully determine invariance.

Contribution

It provides a classification of invariant rank two vector bundles under specific subgroup actions and shows the jumping locus does not uniquely characterize invariance.

Findings

Classified invariant rank two bundles under parabolic subgroup actions.

Discovered infinite families that are almost uniform but not almost homogeneous.

Showed the jumping locus geometry does not determine invariance.

Abstract

In this paper we characterize the rank two vector bundles on $\mathbb{P}^2$ which are invariant under the actions of the parabolic subgroups $G_p:=\mathrm{Stab}_p(\mathrm{PGL}(3))$ fixing a point in the projective plane, $G_L:=\mathrm{Stab}_L(\mathrm{PGL}(3))$ fixing a line, and when $p\in L$, the Borel subgroup $\mathbf{B} = G_p \cap G_L$ of $\mathrm{PGL}(3)$. Moreover, we prove that the geometrical configuration of the jumping locus induced by the invariance does not, on the other hand, characterize the invariance itself. Indeed, we find infinite families that are almost uniform but not almost homogeneous.