Stability and moduli spaces of syzygy bundles

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,\mathrm{.}\mathrm{.}\mathrm{.},d_n}$ on ${\mathbb{P}}^N$ defined as the kernel of a general epimorphism $$\varphi :\mathcal{O}(-d_1)\oplus \mathrm{.}\mathrm{.}\mathrm{.}\oplus \mathcal{O}(-d_n)\to \mathcal{O}$$ is (semi)stable. In this thesis, attention is restricted to the case of syzygy bundles $\mathrm{S}\mathrm{y}\mathrm{z}(f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n)$ on ${\mathbb{P}}^N$ associated to $n$ generic forms $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n\in K[X_0,\mathrm{.}\mathrm{.}\mathrm{.},X_N]$ of the same degree $d$, for $N\ge 2$. The first goal is to prove that $\mathrm{S}\mathrm{y}\mathrm{z}(f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n)$ is stable if $$N+1\le n\le \left(\genfrac{}{}{0px}{}{d+N}{N}\right),$$ 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 $\mathrm{S}\mathrm{y}\mathrm{z}(f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n)$ is unobstructed and it belongs to a generically smooth irreducible component of dimension $n\left(\genfrac{}{}{0px}{}{d+N}{N}\right)-n^2$, if $N\ge 3$, and $n\left(\genfrac{}{}{0px}{}{d+2}{2}\right)+n\left(\genfrac{}{}{0px}{}{d-1}{2}\right)-n^2$, if $N=2$. The results in chapter 3, for $N\ge 3$, were obtained independently by Iustin Coand\u{a} in arXiv:0909.4435.