Syzygy bundles of non-complete linear systems: stability and rigidness

Abstract

Let $(X, L)$ be a polarized smooth projective variety. For any basepoint-free linear system $\mathcal{L}_V$ with $V \subset \mathrm{H}^0\left(X, \mathcal{O}_X(L)\right)$, we consider the syzygy bundle $M_V$ as the kernel of the evaluation map $V \otimes \mathcal{O}_X \rightarrow \mathcal{O}_X(L)$. The purpose of this article is twofold. First, we assume that $M_V$ is $L$-stable and prove that, in a wide family of projective varieties, it represents a smooth point $\left[M_V\right]$ in the corresponding moduli space $\mathcal{M}$. We compute the dimension of the irreducible component of $\mathcal{M}$ passing through $\left[M_V\right]$ and whether it is an isolated point. It turns out that the rigidness of $\left[M_V\right]$ is closely related to the completeness of the linear system $\mathcal{L}_V$. In the second part of the paper, we address a question posed by Brenner regarding the stability of $M_V$ when $V$ is general enough. We answer this question for a large family of polarizations of $X=\mathbb{P}^m \times \mathbb{P}^n$.