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Is there a characterization of projective varieties $X\subset\mathbb{P}^n$ whose secant variety is a hypersurface of degree $3$?

In the case that the secant variety does not have the expected dimension $2\dim(X)+1$, then these are exactly the Severi varieties which are characterized. But there are more examples like for instance the rational normal curve of degree $4$. Is there a complete classification?

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  • $\begingroup$ The rational normal curve is a hyperplane section of the Veronese surface, which is a Severi variety. $\endgroup$
    – Sasha
    Commented Sep 21, 2022 at 21:24
  • $\begingroup$ Yes, indeed. Any thoughts on whether every such variety is the intersection of Severi variety with a linear space? $\endgroup$
    – Hans
    Commented Sep 22, 2022 at 8:27

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