Stability and moduli spaces of syzygy bundles

TL;DR

This paper investigates the stability of syzygy bundles on projective spaces, proving stability under certain conditions and analyzing their moduli spaces, with new results on unobstructedness and irreducible components.

Contribution

It establishes stability criteria for syzygy bundles associated to generic forms and studies their moduli spaces, including unobstructedness and dimension, extending prior work.

Findings

Syzygy bundles are stable for specified ranges of n, d, N.

Syzygy bundles are unobstructed and belong to smooth irreducible components of moduli spaces.

Results include explicit dimension formulas for these components.

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.