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 \[ϕ:\mathcal{O}(-d_1)\oplus...\oplus\mathcal{O}(-d_n) \to\mathcal{O}\] 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 $N\ge 2$. 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{\'ı}a Miró-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 $n\left(\genfrac{}{}{0ex}{}{d+N}{N}\right)-n^2$, if $N\ge 3$, and $n\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)+n\left(\genfrac{}{}{0ex}{}{d-1}{2}\right)-n^2$, if $N=2$. The results in chapter 3, for $N\ge 3$, were obtained independently by Iustin Coandă in arXiv:0909.4435.

PhD thesis

Cited by in corpus (1)

• On the stability of syzygy bundles