Stable sheaves of rank 2 on a 3-dimensional rational scroll (Q810100)

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.