Birational geometry of defective varieties, II

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.
Comment: Accepted for publication in Communications in Algebra