Surfaces in P4 with a family of plane curves

We prove that if S⊆P4 is a smooth nondegenerate surface covered by a one-dimensional family D={Dx}x∈TD={Dx}x∈T of plane (nondegenerate) curves, not forming a fibration, and if the hypersurface given by the union of the planes 〈Dx〉〈Dx〉 spanned by such curves is not a cone, then for any general x∈Tx∈T, the genus g(Dx)≤1g(Dx)≤1, and SS is either: 1.the projected Veronese surface, and the plane curves are conics;2.the rational normal cubic scroll, and the plane curves are conics;3.a quintic elliptic scroll, and the plane curves are smooth cubics. Furthermore, if the number of curves of the family passing through a general point of SS is m≥3m≥3, only cases 11 and 22 may occur.The statement has been conjectured by Sierra and Tironi in [J. Sierra, A. Tironi, Some remarks on surfaces in P4 containing a family of plane curves, J. Pure Appl. Algebra 209 (2) (2007) 361–369., Conjecture 4.13]