A family of rank-2 mathematical instanton bundles on \(\mathbb{P}_3\) (Q1387735)

Let \(M_{2}(0,n)\) denote the coarse moduli scheme of stable rank-2 vector bundles on the projective space \(\mathbb{P}_{3}\) where the Chern classes satisfy \(c_{1}(E) = 0\) and \(c_{2}(E) = n\). Suppose that \(E\) comes from \(M_{2}(0,n)\) and the maps \(f: X\to \mathbb{P}_{3}\), \(g: X\to \mathbb{P}_{2}\) arise from blowing up \(P_{3}\) at some point \(x\in \mathbb{P}_{3}\). If \(\sigma = g^{*}\mathcal{O}(1)\) and \(\tau = f^{*}\mathcal{O}(1)\), then \(\theta = R^{1}g_{*}f^{*}E(-\sigma,-\tau)\) is a sheaf on \(\mathbb{P}_{2}\) with support a curve \(\mathcal{C}\) called a spectral curve of \(E\). Furthermore, \(\theta\) is called a theta characteristic on \(\mathcal{C}\). This means that \(\theta^{\otimes^{2}}= \omega_{\mathcal{C}}\). For such a situation, the main result of the paper, which follows, is used. Theorem. Let \(\mathcal{C}\) be any non-singular plane curve of degree \(n\) and \(\theta\) a theta characteristic on it. A pencil in the projective space \(\mathbb{P} H^{0}\theta (2)\) defines an exact sequence \[ 0\to F\to \mathcal{O}^{2}\to\theta (2)\to 0 \] with \(F\) a rank \(2\) vector bundle and where \(F_{\mathcal{C}}^{{\scriptscriptstyle \vee}}= \theta (1)\oplus\theta (2)\) on the curve \(\mathcal{C}\). As a consequence, the author is able to obtain a special family of rank-2 mathematical instanton bundles on \(\mathbb{P}_{3}\).