Triple-point defective surfaces

Abstract

Abstract In this paper we study the linear series |L – 3p| of hyperplane sections with a triple point p on a surface S embedded via a very ample line bundle L for a general point p. If this linear series does not have the expected dimension we call (S, L) triple-point defective. We show that on a triple-point defective surface through a general point every hyperplane section has either a triple component or the surface is rationally ruled and the hyperplane section contains twice a fibre of the ruling.