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