Triple solids and scrolls (Q6623826)

The question of the classification of smooth triple covers \(\phi: Y \to \mathbb{P}^n\) is of interest, specially when \(n =2,3\), where \(Y\) is not necessarily contained in the total space of an ample line bundle on \(\mathbb{P}^n\) as a triple section. \N\NThe main goal of the paper under review is to study such covers under the additional hypothesis on \(Y\) to have a scroll structure for some polarization. As a consequence of a Barth type result (see Lemma 2.1 and references therein), any \(d\)-uple cover of \(\mathbb{P}^n\) which is also a scroll must satisfy \(d \geq n \geq 2\). Moreover, the authors show that the structure of scroll with respect to some polarization is reflected in a structure of scroll with respect to \(\phi^*\mathcal{O}_{\mathbb{P}^n}(1)\). This leads (see Proposition 3.1) to \((Y, \phi^*\mathcal{O}_{\mathbb{P}^n}(1))=(\mathbb{P}^1\times\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(1,1))\), when \(d=2\). For \(d=3\) and \(Y\) a scroll over a curve (see Theorem 3.2), three different situations appear: (1) \(Y\) is the Segre product of \(\mathbb{P}^2 \times \mathbb{P}^1\); (2) \(Y\) is the Segre-Hirzebruch surface \(\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(-1))\), polarized by the sum of the \((-1)\)-section and a fiber; or (3) \(Y\) is a projective indescomposable bundle \(\mathbb{P}_C(\mathcal{U})\) of degree one over an elliptic curve \(C\) and the polarization is the tautological bundle of the tensor of \(\mathcal{U}\) and \(\mathcal{O}_C(z)\), \(z \in C\). \N\NSections 4 to 6 are devoted to the study of the case \(Y\) a scroll over a surface. In particular, in Section 4 several situations leading to \(Y=\mathbb{P}^2 \times \mathbb{P}^1\) are studied, for instance \(Y\) Fano. The restrictions imposed by the fact that the general element \(S \in |\phi^*\mathcal{O}_{\mathbb{P}^3}(1)|\) is a double plane are studied in Section 5. When the base surface of the scroll structure is a plane, several important restrictions on the numerical invariants are provided, see Section 6. Furthermore, for \(\varphi:S \to \mathbb{P}^2\) general (no total ramification in codimension one and the only singular points of the branch locus are ordinary cusps), \(Y=\mathbb{P}^2 \times \mathbb{P}^1\).