Stability and unobstructedness of syzygy bundles

Abstract

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,\dots ,d_n}$ on ${\mathbb{P}}^N$ defined as the kernel of a general epimorphism

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is (semi)stable. In this note we restrict our attention to the case of syzygy bundles $E_{d,n}$ on ${\mathbb{P}}^N$ associated to $n$ generic forms $f_1,\dots ,f_n\in K[X_0,X_1,\dots ,X_N]$ of the same degree $d$. Our first goal is to prove that $E_{d,n}$ is stable if $N+1\le n\le \left(\genfrac{}{}{0ex}{}{d+2}{2}\right)+N-2$ and $(N,n,d)\ne (2,5,2)$. This bound improves, in general, the bound $n\le d(N+1)$ 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 ${\mathbb{P}}^N$ containing syzygy bundles. We prove that if $N+1\le n\le \left(\genfrac{}{}{0ex}{}{d+2}{2}\right)+N-2$, $N\ne 3$ and $(N,n,d)\ne (2,5,2)$, then the syzygy bundle $E_{d,n}$ is unobstructed and it belongs to a generically smooth irreducible component of dimension $n\left(\genfrac{}{}{0ex}{}{d+N}{N}\right)-n^2$, if $N\ge 4$, and $n\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)+n\left(\genfrac{}{}{0ex}{}{d-1}{2}\right)-n^2$, if $N=2$.