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 $V\otimes \mathcal{O}_{X}\rightarrow \mathcal{O}_{X}(L)$. 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 $[M_{V}]$ 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 $X=\mathbb{P}^{m}\times\mathbb{P}^{n}$.