2023/06/11 by Rosa M. Miró‐Roig, Miró-Roig, Rosa M., Martí Salat-Moltó +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2306.06713
openalex publication_date 2023/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Let (X,L) be a polarized smooth projective variety. For any basepoint-free linear system LV with V⊂ H0(X,OX(L)) we consider the syzygy bundle MV as the kernel of the evaluation map V⊗ OX→ OX(L). The purpose of this article is twofold. First, we assume that MV is L-stable and prove that, in a wide family of projective varieties, it represents a smooth point [MV] in the corresponding moduli space M. We compute the dimension of the irreducible component of M passing through [MV] and whether it is an isolated point. It turns out that the rigidness of [MV] is closely related to the completeness of the linear system LV. In the second part of the paper, we address a question posed by Brenner regarding the stability of MV when V is general enough. We answer this question for a large family of polarizations of X=ℙm×ℙn.