Threefolds with one apparent double point

TL;DR

This paper studies three-dimensional algebraic varieties with exactly one apparent double point, extending previous classifications to cases where the fundamental surface is not a plane, providing partial classification results and new examples.

Contribution

It advances the classification of OADP threefolds by exploring cases with non-planar fundamental surfaces, offering partial results and new examples.

Findings

Partial classification of non-planar fundamental surface cases

Examples of threefolds with one apparent double point

Extension of previous classification results

Abstract

The number of apparent double points of an irreducible projective variety $X$ of dimension $n$ in $\mathbb{P}^{2n+1}$ is the number of secant lines to $X$ passing through a general point of $\mathbb{P}^{2n+1}$. This classical notion dates back to Severi. Smooth threefolds having one apparent double point (shortly OADP threefolds) have been classified in 2004 by Ciliberto, Mella and Russo. In 2011, Ciliberto and Russo have classified normal OADP threefolds such that the so-called fundamental surface is a plane. In this thesis the case in which the fundamental surface is not a plane is considered. A partial classification is given and several examples are presented.