Certain relative quadric hypersurfaces in 3-dimensional projective space bundles over a projective curve
Let \(C\) be either \(\mathbb{P}_{\mathbb{C}}^1\) or a smooth elliptic curve defined over the complex numbers. This paper classifies projective bundles \(f : \mathbb{P}(E) \rightarrow C\), where \(E\) is a rank 4 locally free sheaf on \(C\), such that there exists a divisor \(Q \subset \mathbb{P}(E)\) such that the fibers of the restriction \(Q \rightarrow C\) are isomorphic to \(\mathbb{P}^1\times \mathbb{P}^1\). The classification starts from the observation of K. Konno that if such \(Q\) exists, then \(Q\) belongs to the linear system \(|\mathcal{O}_{\mathbb{P}(E)/C}(2)\otimes f^{\ast}L|\), where \(L\) is an invertible sheaf on \(C\) such that \(E^{\ast} \cong E \otimes L\). Then the author uses the classification of vector bundles on \(\mathbb{P}^1_{\mathbb{C}}\) and elliptic curves in order to identify all such vector bundles \(E\) and \(L\). Then he proceeds by a case by case study. In particular, if \(C = \mathbb{P}^1\), he shows that \(E = \mathcal{O}_{\mathbb{P}^1}(d_1+d_2)\oplus \mathcal{O}_{\mathbb{P}^1}(d_1)\oplus \mathcal{O}_{\mathbb{P}^1}(d_2)\oplus \mathcal{O}_{\mathbb{P}^1}\), \(d_1, d_2 \geq 0\), \(L=\mathcal{O}_{\mathbb{P}^1}(-d_1-d_2)\) and the general member \(Q\) of the linear system \(|\mathcal{O}_{\mathbb{P}(E)/C}(2)\otimes f^{\ast}L|\) is isomorphic to the fiber product \(\mathbb{F}_{d_1} \times_{\mathbb{P}^1} \mathbb{F}_{d_2}\), where \(\mathbb{F}_{d_i}=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P^1}}(d_i) \oplus \mathcal{O}_{\mathbb{P}^1})\). Similar results are presented if \(C\) is a smooth elliptic curve.
scientific article; zbMATH DE number 5956858
scientific article