Moduli spaces of linear bundles on hyperquadrics

In this paper, we will deal with semistable rank 2l   vector bundles on odd dimensional hyperquadrics Q2l+1⊂P2l+2Q2l+1⊂P2l+2 given as the cohomology bundles of linear monads. Firstly we prove that symplectic special linear bundles are stable. Then, inside the Maruyama scheme we consider the locus MLQ2l+1(k)MLQ2l+1(k) parameterizing rank 2l   linear bundles on Q2l+1Q2l+1 with second Chern class k   and we analyze its geometric properties. We prove that the moduli space MLQ2l+1(1)MLQ2l+1(1) is smooth, irreducible of dimension 2l2+5l+22l2+5l+2 and that for any l≥2l≥2 and k≥2k≥2, the moduli space MLQ2l+1(k)MLQ2l+1(k) is singular.