Stability of Quadric bundles

TL;DR

This paper proves that for decorated vector bundles of type (a,b,c,N) on smooth projective curves, checking semistability can be simplified to filtrations of length at most two when a=2, making the stability analysis more manageable.

Contribution

The authors show that for a=2, semistability of decorated bundles can be verified using only filtrations of length ≤ 2, simplifying the stability criterion and providing an explicit destabilizing filtration algorithm.

Findings

Semistability check reduces to filtrations of length ≤ 2 for a=2.

Simplified stability condition coincides with that of orthogonal bundles.

Method extends to decorated bundles on nodal curves.

Abstract

Let $(E,\varphi)$ be decorated vector bundle of type $(a,b,c,N)$ on a smooth projective curve $X$. There is a suitable semistability condition for such objects which has to be checked for any weighted filtration of $E$. We prove, at least when $a=2$, that it is enough to consider filtrations of length less or equal than two. In this case decorated bundles are very close to quadric bundles and to check semistability condition one can just consider the former. A similar result for L-twisted bundles and quadric bundles was already proved. Our proof provides an explicit algorithm which requires a destabilizing filtration and ensures a destabilizing subfiltration of length at most two. Quadric bundles can be thought as a generalization of orthogonal bundles. We show that the simplified semistability condition for decorated bundles coincides with the usual semistability condition for orthogonal bundles. Finally we note that our proof can be easily generalized to decorated vector bundles on nodal curves.