Irreducibility of the singular locus of a cubic hypersurface 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.
 A: Edit. I missed the condition that the secant variety should span the hypersurface. I am leaving the example below for any case. I will think about the secant condition.
Original Post.
That is not true, with the possible exception of a few finite fields such as $\mathbb{F}_2$.  Already for $N\geq 3$, the zero scheme of the following degree $3$ polynomial is integral, normal, and has singular locus equal to $\{[1,0,0,\dots,0],[0,1,0,\dots,0]\}$.  For a hypersurface in $\mathbb{P}^N$, if the singular locus has codimension $\geq 3$ in $\mathbb{P}^N$, then automatically the hypersurface is integral and normal.  This follows from Serre's Criterion for normality. $$F(X_0,X_1,X_2,X_3,X_4\dots,X_N) = X_0X_1(X_2+X_3) + $$ $$(a_3X_2+a_2X_3)X_2X_3 + G(X_4,\dots,X_N).$$  Here $a_2$ and $a_3$ are distinct nonzero elements of the field (these exist if the field is not $\mathbb{F}_2$), and $G(X_4,\dots,X_N)$ is a homogeneous degree $3$ polynomial whose critical locus in $\mathbb{A}^{N-3}$ is just the origin, e.g., $X_4^3+\dots+X_N^3$ if the characteristic is not $3$ (there are similar examples in characteristic $3$ except for finitely many finite fields $\mathbb{F}_{3^r}$).
