Stable sheaves of rank 2 on a 3-dimensional rational scroll
type: Item
instance of: scholarly article
title: Stable sheaves of rank 2 on a 3-dimensional rational scroll
zbMATH Open document ID: 0733.14007
author: Toshio Hosoh
published in: Tokyo Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The author and \textit{S. Ishimura} [J. Math. Soc. Japan 37, 557-568 (1985; Zbl 0608.14011)] showed that a family of stable vector bundles of rank 2 on a 3-dimensional rational scroll forms a complement of a dual 3- dimensional rational scroll. It is natural to ask what is its closure in the moduli space of stable sheaves of same rank and same Chern classes. The goal of this paper is to answer this question. Set \(V={\mathcal O}_{{\mathbb{P}}_ 1}(a)\oplus {\mathcal O}_{{\mathbb{P}}_ 1}(b)\oplus {\mathcal O}_{{\mathbb{P}}_ 1}\) with \(a\leq b\leq 0\), \(\pi\) : X\(=X={\mathbb{P}}(V)\to {\mathbb{P}}_ 1\), \(D:= a\) divisor on X such that \(\pi_*{\mathcal O}_ X(D)\cong V\) and \(F:=\pi^{-1}(x)\) where x is a closed point in \({\mathbb{P}}_ 1\). For an integer \(q\geq 1-a\), set \(H:=D+qF\), \(p:=(DH^ 2)\) and \({\mathcal L}:={\mathcal O}_ X(-D+(p+1)F)\). The couple (X,H) is a 3-dimensional rational scroll. Let M be the moduli space of H-stable sheaves of rank 2 on X with \(c_ 1=c_ 1({\mathcal L})\), \(c_ 2=DF\) and \(c_ 3=0\). The main theorem of this work says that M is connected and has two irreducible components \(M_ 0\) and \(M_ 1\), \(\dim (M_ 1)=\dim (M_ 0)+4\) and \(M_ 0\setminus M_ 1\) consists of all vector bundles.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 4212214
zbMATH Keywords: connectivity of moduli space
zbMATH Keywords: 3-dimensional rational scroll
zbMATH Keywords: moduli space of stable sheaves
zbMATH Keywords: Chern classes
reviewed by: Rosa M. Miró-Roig
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270132264
OpenAlex ID: W2023297388
DOI: 10.3836/TJM/1270132264
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description: scientific article; zbMATH DE number 4212214