Stability and unobstructedness of syzygy bundles

Abstract: It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle on defined as the kernel of a general epimorphism is (semi)stable. In this note we restrict our attention to the case of syzygy bundles on associated to generic forms of the same degree . Our first goal is to prove that is stable if and . This bound improves, in general, the bound given by Hein (2008 ), Appendix A. Display Omitted In the last part of the paper, we study moduli spaces of stable rank vector bundles on containing syzygy bundles. We prove that if , and , then the syzygy bundle is unobstructed and it belongs to a generically smooth irreducible component of dimension , if , and , if . [Copyright &y& Elsevier]