Triple-Point Defective Regular Surfaces

TL;DR

This paper investigates the properties of regular surfaces with very ample line bundles where certain linear series with triple points are defective, revealing geometric structures like ruling or special components in hyperplane sections.

Contribution

It characterizes triple-point defective regular surfaces, showing they are either rationally ruled or have hyperplane sections with specific multiple components.

Findings

Triple-point defectiveness implies special geometric structures.

Regular surfaces with defective linear series are either ruled or contain double fibers.

The study provides a classification of such surfaces based on their hyperplane sections.

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.