Stability and unobstructedness of syzygy bundles

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle Ed1,…,dnEd1,…,dn on PNPN defined as the kernel of a general epimorphism is (semi)stable. In this note we restrict our attention to the case of syzygy bundles Ed,nEd,n on PNPN associated to nn generic forms f1,…,fn∈K[X0,X1,…,XN]f1,…,fn∈K[X0,X1,…,XN] of the same degree dd. Our first goal is to prove that Ed,nEd,n is stable if N+1≤n≤(d+22)+N−2 and (N,n,d)≠(2,5,2)(N,n,d)≠(2,5,2). This bound improves, in general, the bound n≤d(N+1)n≤d(N+1) given by Hein (2008 [2]), Appendix A.In the last part of the paper, we study moduli spaces of stable rank n−1n−1 vector bundles on PNPN containing syzygy bundles. We prove that if N+1≤n≤(d+22)+N−2, N≠3N≠3 and (N,n,d)≠(2,5,2)(N,n,d)≠(2,5,2), then the syzygy bundle Ed,nEd,n is unobstructed and it belongs to a generically smooth irreducible component of dimension n(d+NN)−n2, if N≥4N≥4, and n(d+22)+n(d−12)−n2, if N=2N=2.