Compactification of \(M_{\mathbb{P}_ 3}(0,2)\) and Poncelet pairs of conics (Q1175110)

\textit{R. Hartshorne} [Math. Ann. 238, 229-280 (1978; Zbl 0411.14002)] studied the space \(M(0,2)\) of stable rank 2 vector bundles on \(\mathbb{P}^ 3\) with \(c_ 1=0\), \(c_ 2=2\). In particular, he showed that \(M(0,2)\) is a bundle over a 9-dimensional variety of reguli \(R\) the fibre of which is an open subset of a smooth quadric in \(\mathbb{P}^ 5\). In the paper under review the authors study a natural compactification of \(M(0,2)\) defined as a Poncelet quadric bundle \(Q\) over a natural compactification \(C(\mathbb{G})\) of the variety \(R\). The space \(Q\) essentially parametrizes a family of semi-stable sheaves of rank 2 with \(c_ 1=c_ 3=0\), \(c_ 2=2\), and the smooth points of \(Q\) correspond exactly to stable sheaves. The authors study the relationship between \(Q\) and the closure \(\overline{M(0,2)}\) of \(M(0,2)\) in the Maruyama scheme of semi-stable sheaves on \(\mathbb{P}^ 3\) with \(c_ 1=0\), \(c_ 2=2\), \(c_ 3=0\) and describe explicitly the sheaves on \(\mathbb{P}^ 3\) which occur in the boundary.