Codimension of the non-Lefschetz locus of conics for general semistable rank-2 bundles with even first Chern class

Establish that for a general semistable rank-2 vector bundle E on P^2 with even first Chern class, the non-Lefschetz locus of conics for the module M = H^1_*(P^2, E) has the expected codimension in P^5, namely codimension 2 when E is semistable but not stable and codimension 3 when E is stable.

Background

The paper studies the non-Lefschetz locus of conics—degree-2 forms C for which multiplication ×C fails to have maximal rank in some degree—both for complete intersections and for first cohomology modules of rank-2 vector bundles on P^2. For semistable bundles with odd first Chern class, the locus is a hypersurface and coincides with jumping conics. However, for semistable bundles with even first Chern class, the non-Lefschetz locus is a proper subset of the jumping conics and is expected to have higher codimension.

In Section 5.1, the authors formulate an explicit expected codimension (2 for semistable but not stable, 3 for stable) and conjecture that this is achieved for general E. They later prove this dimension for the special case where E is the syzygy bundle of a general height-3 complete intersection.