Birational geometry of defective varieties, II

TL;DR

This paper investigates the relationship between contact and secant defects of smooth irreducible varieties in projective space, characterizing cases where these defects are equal and analyzing the structure of the tangential contact locus.

Contribution

It establishes an inequality between contact and secant defects for all k and characterizes varieties where equality holds, including singularity and reducibility conditions.

Findings

Proves $ u_k(X) \,\geq\, \delta_k(X)$ for all k.

Characterizes varieties with $ u_1(X) = \delta_1(X)$.

Analyzes the structure of the tangential contact locus in these cases.

Abstract

Let $X \subset \mathbb{P}^r$ be smooth and irreducible and for $k \ge 0$ let $\nu_k(X)$ (resp., $\delta_k(X)$) be the $k$-th contact (resp., the $k$-th secant) defect of $X$. For all $k \ge 0$ we have the inequality $\nu_k(X) \ge \delta_k(X)$ and in the case $k=1$ we characterize projective varieties $X$ for which equality holds, $\dim \mathrm{Sing}(X) \le \delta _1(X) -1$ and the generic tangential contact locus is reducible.