Let $X\subset\mathbb{P}^N$ be a rational smooth projective irreducible non degenerated variety of dimension $n=\dim(X)$ and let $$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^2}|$$ be the linear system of hyperplane sections singular at $p_1,\dots,p_{l}$.

Assume that $\dim(\mathcal{H})=n$ and that the general divisor $H\in \mathcal{H}$ is singular along a positive dimensional subvariety passing through the points. Moreover assume that the schematic intersection of all of these singularities is singular only at $p_1,\dots,p_l$.

Locally, in a neighborhood of the $p_i$'s, say $p_1$ we can write $H$ as the zero locus of $$H_f=a_{i,j}x_ix_j+h(x_1,\dots,x_n)=0$$ where $x_1,\dots,x_n$ are local coordinates and $h(x_1,\dots,x_n)$ is a polynomial of degree $\geq3$. The quadric $$Q_{H_f}=a_{i,j} x_ix_j=0$$ have in general rank $h\leq n$. Denote with $\mathcal{A}_H$ the vertex of $Q_{H_f}$. Is it true that under these assumptions $\bigcap_{H\in \mathcal{H}} \mathcal{A}_H=p_1$?