On a family of instanton bundles (Q1365447)

Let \(E\) be an \(n\)-instanton: a theorem of Grothendieck asserts that the restriction of \(E\) to a line \(L\) is of the form: \({\mathcal O}_L(a)\oplus {\mathcal O}_L(-a)\). If \(a>0\), \(L\) is said to be the \(a\)-jumping line for \(E\). Let \({\mathcal S}_n\) be the family of \(n\)-instantons with (at least) one \(n\)-jumping line, \({\mathcal S}_n\) coincides with the moduli space of \(n\)-instantons for \(n=1\) or 2, and it is known that it is an irreducible rational variety of dimension \(8n-3\). Here it is shown that any element \(E\) on \({\mathcal S}_n\) is an extension (by Serre construction) from a curve with support \(L\), which is a ``double structure'' on the infinitesimal neighborhood of \(L\) of order \(n\). The monadics description deduced from this construction gives a certain morphism from an Zariski-open subset of \(\text{Proj}_G({\mathcal A})\) to \({\mathcal S}_n\) where \(G=\text{Gr}(1,3)\) and \({\mathcal A}\) is a vector bundle on \(G\). In particular \({\mathcal S}_n\) is an irreducible rational variety of dimension \(6n+2\) for \(n\geq 3\).