Mathematical instantons with maximal order jumping lines
A mathematical instanton bundle on \(\mathbb{P}^3\) (over an algebraically closed field) is a rank two vector bundle \({\mathcal E}\) on \(\mathbb{P}^3\) with \(c_1= 0\) and with \(h^0 ({\mathcal E})= H^1 ({\mathcal E} (-2)) =0\). Let \(c_2 ({\mathcal E})= n\). Then \(n> 0\). A jumping line of \({\mathcal E}\) of order \(a\) \((a>0)\), is a line \(\ell\) in \(\mathbb{P}^3\) on which \({\mathcal E}\) splits as \({\mathcal O}_\ell (-a) \oplus {\mathcal O}_\ell (a)\). It is easy to see that the jumping lines of \({\mathcal E}\) all have order \(\leq n\). We will say that \({\mathcal E}\) has a maximal order jumping line if it has a jumping line of order \(n\). Our goal is to show that such an \({\mathcal E}\) is unobstructed in the moduli space of stable rank two bundles, i.e., \(H^2 ({\mathcal E} \otimes {\mathcal E}) =0\). The technique can be slightly extended. We show that when \(c_2= 5\), any \({\mathcal E}\) with a jumping line of order 4 is unobstructed. We describe at the end how mathematical instantons with maximal order jumping lines arise and estimate the dimension of this particular smooth locus of bundles.
scientific article; zbMATH DE number 950218