On the singularities of the trisecant surface to a space curve

Abstract

Let C be a smooth curve in P^3 . Trisecant lines to the curve C are, in general, sweeping out a (reduced) surface in P^3 . In this note we attempt to describe some of its singularities, and in particular we show that if the curve C has only a finite number of quadrisecant lines, then the singular locus contains the quadrisecant lines to C, and the points through which pass more trisecants than through a generic point of the trisecant surface. Several explicit examples are discussed in the last section.