Stability and Unobstructedness of Syzygy Bundles

TL;DR

This paper investigates the stability and unobstructedness of syzygy bundles on projective space, providing improved bounds for stability and analyzing their moduli space properties, including smoothness and dimension.

Contribution

It proves stability of certain syzygy bundles under new bounds and demonstrates their unobstructedness and moduli space characteristics for a range of parameters.

Findings

Syzygy bundles are stable if n satisfies the new bounds.

Syzygy bundles are unobstructed and belong to smooth moduli components.

The dimension of the moduli space components is explicitly computed.

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 $\xymatrix{\phi:\cO(-d_1)\oplus...\oplus\cO(-d_n)\ar[r] &\cO}$ 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 $N+1\le n\le\tbinom{d+2}{2}+N-2$. This bound improves, in general, the bound $n\le d(N+1)$ 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 $N+1\le n\le\tbinom{d+2}{2}+N-2$ and $N\ne 3$, then the syzygy bundle $E_{d,n}$ is unobstructed and it belongs to a generically smooth irreducible component of dimension $n\tbinom{d+N}{N}-n^2$, if $N \geq 4$, and $n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2$, if N=2.