On the classification of defective threefolds

TL;DR

This paper classifies all irreducible projective threefolds that are defective with respect to their secant varieties, extending classical results for the case when k=1, and provides a comprehensive understanding of their geometric properties.

Contribution

It extends Scorza's classical classification to all k-defective irreducible projective threefolds, offering a complete classification in this broader context.

Findings

Complete classification of k-defective threefolds

Extension of classical results for k=1

Identification of geometric properties of defective threefolds

Abstract

We classify all irreducible projective threefolds $X$ which are $k$-defective, i.e. some $k$-secant variety of $X$ has dimension less than the expected value. This results extends the classical Scorza's classification of the case $k=1$.