We prove that if is a smooth nondegenerate surface covered by a one-dimensional family D={Dx}x∈T of plane (nondegenerate) curves, not forming a fibration, and if the hypersurface given by the union of the planes 〈Dx〉 spanned by such curves is not a cone, then for any general x∈T, the genus g(Dx)≤1, and S 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 S is m≥3, only cases 1 and 2 may occur.
The statement has been conjectured by Sierra and Tironi in [J. Sierra, A. Tironi, Some remarks on surfaces in containing a family of plane curves, J. Pure Appl. Algebra 209 (2) (2007) 361–369., Conjecture 4.13]
