Stability and unobstructedness of syzygy bundles

Abstract

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle E d 1 , … , d n on P N defined as the kernel of a general epimorphism [Display omitted] is (semi)stable. In this note we restrict our attention to the case of syzygy bundles E d , n on P N associated to n generic forms f 1 , … , f n ∈ K [ X 0 , X 1 , … , X N ] of the same degree d . Our first goal is to prove that E d , n is stable if N + 1 ≤ n ≤ ( d + 2 2 ) + N − 2 and ( N , n , d ) ≠ ( 2 , 5 , 2 ) . This bound improves, in general, the bound 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 − 1 vector bundles on P N containing syzygy bundles. We prove that if N + 1 ≤ n ≤ ( d + 2 2 ) + N − 2 , N ≠ 3 and ( N , n , d ) ≠ ( 2 , 5 , 2 ) , then the syzygy bundle E d , n is unobstructed and it belongs to a generically smooth irreducible component of dimension n ( d + N N ) − n 2 , if N ≥ 4 , and n ( d + 2 2 ) + n ( d − 1 2 ) − n 2 , if N = 2 .