Classification of secant defective manifolds near the extremal case (Q2862168)

Secant defective manifolds are projective manifolds \(X\subset\mathbb P^N\) of dimension \(n\), whose variety of secant lines has dimension smaller than the expected value \(\min\{N,2n+1\}\). It is well known that such varieties span a space of dimension at most \(N=\binom{n+1}2-1\), and the equality is attained only by the \(2\)-Veronese embeddings \(v_2(\mathbb P^n)\). Quasi-extremal cases, where \(N=\binom{n+2}2-2\), were classified by \textit{R. Muñoz} et al. [Bull. Lond. Math. Soc. 39, No. 6, 949--961 (2007; Zbl 1143.14041)].NEWLINENEWLINEThe author extends the description of secant defective manifolds to the case where \(N\geq \binom{n+2}2-(n-2).\) Secant defective manifolds in the range are proven to be varieties with quadratic entry locus, a type deeply studied by Ionesco and Russo. The author deduces that when \(N\geq \binom{n+2}2-(n-2)\), secant defective manifolds of dimension \(n\geq 2\) are isomorphic projections either of \(v_2(\mathbb P^n)\), or of its projections from the span of the image of a subspace \(\mathbb P^s\subset\mathbb P^n\), with \(\binom{s+2}2\leq n-2\).