# Singular locus of a linear system of hyperplane sections

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 $$\bigcap_{H\in \mathcal{H}} \mathcal{A}_H=p_1$$?

I have the feeling this question is not well-posed. Each local system where $$H$$ is of the form: $$a_{j,i} x_ix_j + h(x_1, \ldots, x_n)$$ depends on $$H$$. As a consequence, $$\mathcal{A}_H$$ lives in a local system that depends on $$H$$. So taking the intersection for all $$H \in \mathcal{H}$$ is somehow a nonsense, since all these linear spaces live in different neighborhoods.
What could be more meaningful would be to define $$\bigcap_{H \in \mathcal{H}} \mathcal{A}_H$$ inside $$T_{X,x}$$, but that would require to write everything in terms of second fundamental forms...
Anyways, even with this reformulation, the question you ask has a negative answer. Let $$X = \mathbb{P}^{6} \times \mathbb{P}^{6} \subset \mathbb{P}^{48}$$. Let $$S^2(X)$$ be the variety of trisecant planes to $$X$$. We have $$\dim S^2(X) = \dim S(X) + X + 1 -4 = 32$$. Let $$x_1,x_2,x_3$$ be generic point in $$X$$ and let $$L = \langle x_1,x_2,x_3 \rangle$$. Let $$y$$ be a generic point in $$L$$ and let $$S_y$$ be the entry locus in $$X$$ associated to $$y$$. This entry locus is a $$\mathbb{P}^2 \times \mathbb{P}^2$$ containing $$x_1,x_2,x_3$$.
By Terracini's lemma, we have $$T_{X,x} \subset T_{S^2(X),y}$$ for all $$x \in S_y$$. As a consequence, any hyperplane in the linear system $$T_{S^2(X),y}^{\perp}$$ is tangent to $$X$$ along $$S_y$$.
Now, let $$\mathcal{H} = T_{S^2(X),y}^{\perp}$$, we have $$\mathcal{H} = |\mathcal{O}_{X}(1) \otimes \mathcal{I}_{x_1,x_2,x_3}^2|$$ and $$\dim \mathcal{H} = 15$$. The generic $$H \in \mathcal{H}$$ is tangent to $$X$$ along $$S_y$$, hence all $$H \in \mathcal{H}$$ are tangent to $$X$$ at least along $$S_y$$. From the definition of $$\mathcal{A}_H$$ (which naturally leaves in $$T_{X,x}$$), we see that $$T_{Sy,x} \subset \mathcal{A}_H$$, for all $$H \in \mathcal{H}$$.