Irreducibility and singularities of some nested Hilbert schemes

2021/06/11 by Tim Ryan, Ryan, Tim, Gregory B. Taylor +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2106.06570

openalex publication_date 2021/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S be a smooth projective surface over ℂ. We study the local and global geometry of the nested Hilbert scheme of points S^[n,n+1,n+2]. In particular, we show that S^[n,n+1,n+2] is an irreducible local complete intersection with klt singularities. In addition, we compute the Picard group of S^[n,n+1,n+2] when h¹(S,O_S) = 0. From the irreducibility of S^[n,n+1,n+2], we deduce irreducibility for four other infinite families of nested Hilbert schemes. We give the first explicit example of a reducible nested Hilbert scheme, which allows us to show that S^{[n₁,…,n_k]} is reducible for k > 22.

Related

• Irreducibility of Some Nested Hilbert Schemes
• Invariants of nested Hilbert and Quot schemes on surfaces
• Hilbert schemes of points on surfaces
• The Hilbert schemes of points on surfaces with rational double point singularities
• Unexpected but recurrent phenomena for Quot and Hilbert schemes of points