Stability and Unobstructedness of Syzygy Bundles

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle Ed1,...,dn on P N defined as the kernel of a general epimorphism φ : O(−d1)⊕ · · · ⊕ O(−dn) O is (semi)stable. In this note, we restrict our attention to the case of syzygy bundles Ed,n on P associated to n generic forms f1, . . . , fn ∈ K[X0, X1, . . . , XN ] of the same degree d. Our first goal is to prove that Ed,n is stable if N + 1 ≤ n ≤ ( d+2 2 ) + N − 2 and (N,n, d) 6= (2, 5, 2). This bound improves, in general, the bound n ≤ d(N + 1) given by G. Hein in [2], Appendix A. In the last part of the paper, we study moduli spaces of stable rank n−1 vector bundles on P containing syzygy bundles. We prove that if N + 1 ≤ n ≤ ( d+2 2 ) +N − 2, N 6= 3 and (N,n, d) 6= (2, 5, 2), then the syzygy bundle Ed,n is unobstructed and it belongs to a generically smooth irreducible component of dimension n ( d+N N ) − n, if N ≥ 4, and n ( d+2 2 )