On a theorem of Severi

Abstract

P 2r+1, on the projection we will obtain a finite number δ(X) of improper nodes, i.e. non-normal double points whose tangent cones break into two r-planes. The number δ(X) is called the number of apparent double points ofX and it is equal to the number of secant lines to X going through a general point of P2r+1. If X ⊂ P3 is a non-degenerate curve, then it has necessarily apparent double points because the secant variety of X fills the whole space and δ(X) is then the difference between the arithmetic genus of the projection of X from a general point and the geometric genus ofX; so ifX has degreed, then δ(X) = (d−1)(d−2) 2 − g(X) > 0. On the other hand it is an immediate consequence of Castelnuovo’ s bound for the genus of a curve that the only smooth curve in P3 having one apparent double point is the rational normal curve of degree 3. In 1901 Severi proved that the only surface in P5 having degenerate secant variety, i.e. the only surface that can be projected isomorphically in P4 (or equivalently without apparent double points), is the Veronese surface in P5. In the same paper Severi claims that the smooth surfaces in P5 having one apparent double point are only the rational normal scroll(s) of degree 4 and the Del Pezzo surface of degree 5 (see [19]). Unfortunately Severi gives a proof which works provided the surface S has not too many trisecants. This gap was remarked first by Ciliberto and Sernesi. Here we prove that the surfaces in P5 having one apparent double point are the rational normal scrolls S(2, 2) and S(1, 3) of degree 4 and the Del Pezzo surface of degree 5.