Surfaces in \(\mathbb{P}^4\) with a family of plane curves (Q1020955)

Consider a smooth projective surface \(S \subset \mathbb{P}^4\) in the four dimensional projective space. Suppose moreover that \(S\) is covered by a one dimensional family \({\mathcal D}=\{D_x\}_{x \in T}\) of plane curves. A natural problem in projective geometry is the classification of such kind of surfaces. When the family forms a fibration the classification has been completed by several authors (see the Introduction of the paper under review for proper references). When the family does not form a fibration, \textit{J. C. Sierra} and \textit{A. L. Tironi} [J. Pure Appl. Algebra 209, No. 2, 361--369 (2007; Zbl 1107.14027)] proved some results confirming the conjecture that the Veronese surface and the quintic elliptic scrolls are the only ones that do not lie on a quadric cone. These authors also stated a conjecture (Conj. 4.13 in the paper quoted above) whose proof is the main result in the paper under review. The precise result provided in this paper is the following (Thm. 0.3). Consider \(S \subset \mathbb{P}^4\) as before such that the curves in \({\mathcal D}\) are not degenerate and do not form a fibration. Let \(m \geq 2\) the number of curves passing through a general point of \(S\). Assume that the hypersurface \(V_T=\cup_{x \in T}\langle D_x\rangle\) given by the planes \(\langle D_x\rangle\) spanned by the curves in \({\mathcal D}\) is not a cone. Then, for the general \(x \in T\), the genus \(g(D_x) \leq 1\) and \(S\) is either the projected Veronese surface, or the rational normal scroll, or a quintic elliptic scroll. Moreover, if \(m \geq 3\) only the first two cases occur.