Syzygy Bundles of Non-complete Linear Systems: Stability and Rigidness

Abstract

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