On the number of Vedernikov-Ein irreducible components of the moduli space of stable rank 2 bundles on the projective space (Q1642306)

Ein bundles are vector bundles of rank 2 on \(\mathbb{P}^3\) generated as cohomology of a monad of the form \[0\to \mathcal{O}(-c)\to \mathcal{O}(-a)\oplus\mathcal{O}(-b)\oplus\mathcal{O}(a)\oplus\mathcal{O}(b)\to \mathcal{O}(c)\to 0\] for \(b\ge a \ge0\) and \(c>a+b\). \textit{L. Ein} [Nagoya Math. J. 111, 13--24 (1988; Zbl 0663.14012)] proved that such bundles are stable and the moduli space \(M(0,c^2-a^2-b^2)\) has an irreducible component in which they are an open dense subset. \textit{V. K. Vedernikov} [Math. USSR, Izv. 25, 301--313 (1985; Zbl 0589.14017); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 5, 986--998 (1984)] considered two special cases of Ein bundles: when \(a=0\) (Vedernikov-Ein components of I type) and when \(a=b\) (Vedernikov-Ein components of II type). In the paper under review, the authors give formulas which compute the number of Vedernikov-Ein components of both types. As a consequence they obtain a criterion for the existence of such components.