Stable vector bundles of \(rank\quad 2\) on a 3-dimensional rational scroll (Q1086304)

This paper deals with 2-bundles on the \({\mathbb{P}}^ 2\)-bundle \(\pi: X={\mathbb{P}}(V)\to {\mathbb{P}}^ 1,\) \(V={\mathcal O}(a)\oplus {\mathcal O}(b)\oplus {\mathcal O}\), \(a\leq b\leq 0\), which are stable with respect to a polarization \(H=D+qF\), where \(\pi_*{\mathcal O}(D)=V\), F a fibre of \(\pi\). The authors give the following nice description of the moduli space of M of such bundles E on X with Chern classes \(c_ 1(E)=-D+(p+1)F\), \(p=2q+a+b\), \(c_ 2(E)=D\cdot F:\) Let \({\tilde \pi}: \tilde X={\mathbb{P}}(V^{\vee})\to {\mathbb{P}}^ 1\) be the 'dual' of X and \(\tilde F\) a fibre of \({\tilde \pi}\), \(\tilde D\) a divisor with \({\tilde \pi}_*{\mathcal O}(\tilde D)=V^{\vee}\). Then \(\tilde H=\tilde D+(p+1)\tilde F\) is very ample on \(\tilde X\) and M becomes isomorphic to the complement of the rational scroll \(\phi_{| \tilde H|}(\tilde X)\) in \({\mathbb{P}}(H^ 0(\tilde X,{\mathcal O}(\tilde H)))\) via the canonical isomorphism \({\mathbb{P}}(Ext^ 1_ X({\mathcal O}(-D+(p+1)F)\), \(\pi^*\omega_{{\mathbb{P}}^ 1})^{\vee})={\mathbb{P}}(H^ 0(\tilde X,{\mathcal O}(\tilde H)))\).