Stability and moduli spaces of syzygy bundles

Abstract

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,...,d_n}$ on $\mathbb{P}^N$ defined as the kernel of a general epimorphism \[\phi:\mathcal{O}(-d_1)\oplus...\oplus\mathcal{O}(-d_n) \to\mathcal{O}\] is (semi)stable. In this thesis, attention is restricted to the case of syzygy bundles $\mathrm{Syz}(f_1,...,f_n)$ on $\mathbb{P}^N$ associated to $n$ generic forms $f_1,...,f_n\in K[X_0,...,X_N]$ of the same degree $d$, for ${N\ge2}$. The first goal is to prove that $\mathrm{Syz}(f_1,...,f_n)$ is stable if \[N+1\le n\le\tbinom{d+N}{N},\] except for the case ${(N,n,d)=(2,5,2)}$. The second is to study moduli spaces of stable rank ${n-1}$ vector bundles on $\mathbb{P}^N$ containing syzygy bundles. In a joint work with Laura Costa and Rosa Mar{\'\i}a Mir\'o-Roig, we prove that $N$, $d$ and $n$ are as above, then the syzygy bundle $\mathrm{Syz}(f_1,...,f_n)$ is unobstructed and it belongs to a generically smooth irreducible component of dimension ${n\tbinom{d+N}{N}-n^2}$, if ${N\ge3}$, and ${n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2}$, if ${N=2}$. The results in chapter 3, for $N\ge3$, were obtained independently by Iustin Coand\u{a} in arXiv:0909.4435.