Birational geometry of defective varieties, II

Abstract

Let $X \subset \mathbb{P}^r$ be smooth and irreducible and for $k \ge 0$ let $ν_k(X)$ (resp., $δ_k(X)$) be the $k$-th contact (resp., the $k$-th secant) defect of $X$. For all $k \ge 0$ we have the inequality $ν_k(X) \ge δ_k(X)$ and in the case $k=1$ we characterize projective varieties $X$ for which equality holds, $\dim \mathrm{Sing}(X) \le δ_1(X) -1$ and the generic tangential contact locus is reducible.