# Triple-Point Defective Surfaces

## TL;DR

This paper investigates the properties of surfaces where the linear series with a triple point does not behave as expected, revealing conditions related to rational rulings or special components in hyperplane sections.

## Contribution

It characterizes triple-point defective surfaces, showing they are either rationally ruled or have hyperplane sections with a triple component.

## Key findings

- Triple-point defectiveness implies special geometric structures.
- Such surfaces are either rationally ruled or contain hyperplane sections with double fibers.
- The study provides criteria for identifying triple-point defective surfaces.

## 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 \emph{general} point $p$. If this linear series does not have the expected dimension we call $(S,L)$ \emph{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.