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 an abstract 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?

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?