Let $Z\subseteq \mathbb{P}^{N}$ be an irreducible cubic hypersurface, i.e. $Z=V(F)$ for certain homogeneous irreducible polynomial $F\in K[X_{0},\ldots,X_{N}]$ of degree $3$. Let us suppose that its singular locus $$ X=V(\frac{\partial F}{\partial X_{0}},\ldots,\frac{\partial F}{\partial X_{N}}) $$ is smooth (in particular $Z$ is not a cone) and that the secant variety of $X$ is $SX=Z$. I would like to prove that in this case $X$ is irreducible.

I don't know if this is true, but at least, I would like to know if there is some easy criterion to check irreducibility under these hypothesis.