Special rank two vector bundles over Enriques surfaces
Let \(\pi : X \to S\) with \(S\) an Enriques surface over \(\mathbb{C}\), \(X\) the K3-universal cover and \(\sigma\) the corresponding involution. A simple rank two vector bundle \(E\) over \(S\) is called special if \(E\simeq E\otimes[K_ S]\), where \([K_ S]\) is the canonical line bundle of \(S\). If \(E\) is special, then there is a line bundle \(L\) over \(X\) which is not \(\sigma\)-invariant such that: \((*)\) \(E = \pi_ * L\). This is the key remark of the paper. In particular the bundle is stable and \(c_ 2 (\text{End} E) = c^ 2_ 1(E) - 4c_ 2(E) = - 2k\), \(k \geq 1\), \(k \in \mathbb{Z}\). For \(k=1\) the bundle is extremal. In the paper it is shown that any extremal rank two vector bundle over an Enriques surface is a Reye bundle, i.e. \(S \subset G(1, \mathbb{P}^ n)\) being homologous to \((3n + 2) \sigma' + n \sigma''\), where \(\sigma'\) and \(\sigma''\) are Schubert cycles, and the extremal bundle is the restriction of the dual of the tautological bundle. The proof is given for special bundles using \((*)\): the line bundle \(L\) modulo tensoring with very ample line bundles, gives an embedding of \(X\) in \(\mathbb{P}^ n\). \(S\) appears as the complex of lines \(\{Sp (x, \sigma (x)) | x \in X \subset \mathbb{P}^ n\}\). The special bundles are shown to represent exactly the singularities of the moduli spaces of stable rank two vector bundles over Enriques surfaces. Furthermore, these singular points are ordinary double points. For the proof one uses again \((*)\). At the end the moduli spaces of dimension one are considered. They are birationally isomorphic with curves of arithmetic genus one.
extremal rank two vector bundle over Enriques surface
scientific article; zbMATH DE number 704907
scientific article