paper

Stability and Unobstructedness of Syzygy Bundles

arXiv:0901.2457 · doi:10.1016/j.jpaa.2009.10.009

Abstract

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_{1}, ..., d_{n}}$ on $\PP^N$ defined as the kernel of a general epimorphism $\xymatrix{Ï:\cO(-d_1)\oplus...\oplus\cO(-d_n)\ar[r] &\cO}$ is (semi)stable. In this note, we restrict our attention to the case of syzygy bundles $E_{d, n}$ on $\PP^N$ associated to $n$ generic forms $f_{1}, ..., f_{n} \in K [X_{0}, X_{1}, ..., X_{N}]$ of the same degree $d$ . Our first goal is to prove that $E_{d, n}$ is stable if $N + 1 \leq n \leq (2 d + 2) + N - 2$ . This bound improves, in general, the bound $n \leq d (N + 1)$ given by G. Hein in \cite{B}, Appendix A. In the last part of the paper, we study moduli spaces of stable rank $n - 1$ vector bundles on $\PP^N$ containing syzygy bundles. We prove that if $N + 1 \leq n \leq (2 d + 2) + N - 2$ and $N \neq = 3$ , then the syzygy bundle $E_{d, n}$ is unobstructed and it belongs to a generically smooth irreducible component of dimension $n (N d + N) - n^{2}$ , if $N \geq 4$ , and $n (2 d + 2) + n (2 d - 1) - n^{2}$ , if N=2.

32 pages, minor changes