Classification of secant defective manifolds near the extremal case

Let $X\subset \P^N$ be a nondegenerate irreducible closed subvariety of dimension $n$ over the field of complex numbers and let $SX\subset\P^N$ be its secant variety. $X\subset\P^N$ is called `secant defective' if $\dim(SX)$ is strictly less than the expected dimension $2n+1$. In \cite{Z1}, F.L. Zak showed that for a secant defective manifold necessarily $N\le{n+2 \choose n}-1$ and that the Veronese variety $v_2(\P^n)$ is the only boundary case. Recently R. Mu$\tilde{\textrm{n}}$oz, J. C. Sierra, and L. E. Sol\'a Conde classified secant defective varieties next to this extremal case in \cite{MSS}. In this paper, we will consider secant defective manifolds $X\subset\P^N$ of dimension $n$ with $N={n+2 \choose n}-1-\epsilon$ for $\epsilon\ge0$. First, we will prove that $X$ is a $LQEL$-manifold of type $\delta=1$ for $\epsilon\le n-2$ (see Theorem \ref{main_thm}) by showing that the tangential behavior of $X$ is good enough to apply Scorza lemma. Then we will completely describe the above manifolds by using the classification of conic-connected manifolds given in \cite{IR1}. Our method generalizes previous results in \cite{Z1,MSS}.