Smooth surfaces with defective secant variety

Question

I am interested in smooth nonedegenerate surfaces $X\subset\mathbb{P}^n$, $n\geq 5$, whose secant variety $\sigma(X)$ has dimension $4$. Clearly, the second Veronese embedding of $\mathbb{P}^2$ is such an example. I would be happy about an answer to any of the following

Questions: Which other examples exist? Is there a characterization of all such surfaces?

Given a projective surface $X$ and a very ample divisor $D$ on $X$, are there convenient criteria on $D$ that guarantee that the secant variety of the embedding of $X$ via the complete linear system $|D|$ has dimension $4$? Maybe in terms of cohomology and/or some intersection product?

Answer 1

The Veronese surface in $\mathbb{P}^5$ is indeed the only secant defective surface that is not a cone. This is a classical (and non-trivial) result by F. Severi, see p. 6 and Theorem 10.1 in

C. Ciliberto, F. Russo: Varieties with minimal secant degree and linear systems of maximal dimension on surfaces, Adv. Math. 200, No. 1, 1-50 (2006). ZBL1086.14043

ad some of the references cited in the Introduction of the same paper, notably [14, 54, 57].