On dual defective manifolds

TL;DR

This paper characterizes dual defective manifolds, establishes bounds on their dual defect, and provides new proofs and conjectures related to their geometric properties, advancing understanding in algebraic geometry.

Contribution

It introduces a new characterization of scrolls among dual defective manifolds and improves bounds on the dual defect, offering a simplified proof of the Landman Parity Theorem.

Findings

Characterization of scrolls among dual defective manifolds

An optimal bound for the dual defect is established

A simplified proof of the Landman Parity Theorem is provided

Abstract

An embedded manifold is dual defective if its dual variety is not a hypersurface. Using the geometry of the variety of lines through a general point, we characterize scrolls among dual defective manifolds. This leads to an optimal bound for the dual defect, which improves results due to Ein. Among other things we also provide a short and easy proof of the famous Landman Parity Theorem for dual defective manifolds based on our approach to the subject. We also discuss our conjecture that every dual defective manifold with cyclic Picard group should be secant defective, of a very special type, namely a local quadratic entry locus variety.