Surfaces in [formula omitted] with a family of plane curves

Abstract

We prove that if S ⊆ P 4 is a smooth nondegenerate surface covered by a one-dimensional family D = { D x } x ∈ T of plane (nondegenerate) curves, not forming a fibration, and if the hypersurface given by the union of the planes 〈 D x 〉 spanned by such curves is not a cone, then for any general x ∈ T , the genus g ( D x ) ≤ 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 P 4 containing a family of plane curves, J. Pure Appl. Algebra 209 (2) (2007) 361–369., Conjecture 4.13]