Stability and moduli spaces of syzygy bundles

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...oplusmathcal{O}(-d_n) tomathcal{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 $f_1,...,f_nin K[X_0,...,X_N]$ of the same degree $d$, for ${Nge2}$. The first goal is to prove that $mathrm{Syz}(f_1,...,f_n)$ is stable if [N+1le nletbinom{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 Miro-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 ${ntbinom{d+N}{N}-n^2}$, if ${Nge3}$, and ${ntbinom{d+2}{2}+ntbinom{d-1}{2}-n^2}$, if ${N=2}$. The results in chapter 3, for $Nge3$, were obtained independently by Iustin Coandu{a} in arXiv:0909.4435.