Consider the Veronese map $\mathbb{P}^{2}\rightarrow\mathbb{P}^{5}$ and let $X\subset\mathbb{P}^{5}$ be the cubic hypersurface defined by $\det(A)=0$. Now $X$ is the image of all reducible conics.
I want to prove that $Sing(X)=\{[A]\subset\mathbb{P}^{5}\mid rk(A)=1\}$, which is indeed the image of reduced conics. What should I do?
Then, as a consequence, I think I would have that if $X\subset\mathbb{P}^{n}$ is a hypersurface of degree $d\geq1$ containing $L$ linear subspace with $dim L\geq\frac{n}{2}$, then $X$ is singular. Which is the statement I initially wanted to prove.
I tried to prove directly the second statement, considering $I=\left\{(p,[F])\mid p\in V\left(F,\dfrac {\partial F}{\partial x_0}, \dots, \dfrac {\partial F}{\partial x_n}\right)\right\}$ and trying to show that $\dim I\geq0$,but I didn't succeed.