Stability of quadric bundles

A decorated vector bundle on a smooth projective curve (X) is a pair ((E,varphi )) consisting of a vector bundle and a morphism (varphi :(E^{otimes a})^{oplus b}rightarrow (det E)^{otimes c}otimes mathsf {N}) , where (mathsf {N}in text {Pic}(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 (le ) 2. 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 Higgs bundles and quadric bundles was already proved (García-Prada et al. in The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations, 2012; Schmitt in Geometric Invariant Theory and Decorated Principal Bundles, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zurich, 2008). 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.