Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$.

Assume that for every $i$ and for every $j$ the polynomial $f$ belongs to the ideal $\langle P_i, Q_j \rangle$.

Is it true that the rank of $\{P_1, \ldots, P_m \}$ (in the vector space of all linear polynomials $\mathbb{C}[x_0,\ldots,x_n]$) or the rank of $\{Q_1, \ldots Q_k \}$ is less than some constant?

(I think the answer of this question can help to solve this question)

**UPD**(First I asked this question for $f=x_0^2$ and Zach Teitler answered to it.)

I think I have understood my question better:).

It is not difficult to see that $f \in \langle P_i, Q_j \rangle \Leftrightarrow$ for every $x$ in subspace $\{t\in \mathbb{C}^n | P_i(t)=Q_j(t)=0\}$, $f(x)=0$.

So, we can ask some another **question**.

Let $M$ be a quadratic surface (the zeros of $f$). Assume that $M$ contains some finite set of subspaces of codimension $2$. What can we say about this set of subspaces? Can this set be large?