Triple-Point Defective Regular Surfaces

Abstract

In this paper we study the linear series |L-3p| of hyperplane sections with a triple point p of 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 regular 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.

The results of this paper have been generalised in the paper Triple point defective surfaces (arXiv:0911.1222) by the same authors. The assumptions on the linear system and on the surface have been weakened. Large parts of the new paper coincide with this old paper and the reader should rather refer to the new paper than to this old one