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

Let $(X,L)$ be a polarized smooth projective variety. For any basepoint-free linear system ${\mathcal{L}}_V$ with $V\subset H^0(X,{\mathcal{O}}_X(L))$ we consider the syzygy bundle $M_V$ as the kernel of the evaluation map $V\otimes {\mathcal{O}}_X\to {\mathcal{O}}_X(L)$. 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 $[M_V]$ in the corresponding moduli space $\mathcal{M}$. We compute the dimension of the irreducible component of $\mathcal{M}$ passing through $[M_V]$ and whether it is an isolated point. It turns out that the rigidness of $[M_V]$ 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 $X={\mathbb{P}}^m\times {\mathbb{P}}^n$.