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?
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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?



Smooth surfaces with defective secant variety
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](http://dx.doi.org/10.1016/j.aim.2004.10.008), *Adv. Math.* **200**, No. 1, 1-50 (2006). [ZBL1086.14043](https://zbmath.org/?q=an:1086.14043)


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


