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](https://mathoverflow.net/questions/286922/sylvester-gallai-type-theorem-for-quadratic-polynomials) 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?



Linear homogenous polynomials that generates one quadratic polynomial
We will assume that $f$ is irreducible (if $f$ is not irreducible then in fact the argument of Zach Teitler's answer works).


Consider $M:= f \cap P\_1$ (I mean the intersection of the zeros $f$ and $P\_1$).


This set is the zeros of a quadratic form in plane $P\_1$ of codimension $1$ (it can not be $P\_1$ since $f$ is irreducible). For every $i$ the intersection of $Q\_i$ and $f$ must contain a subspace of codimension $2$. Hence $M$ is the union of one or two subspace of codimension $2$. So, there exists at most $2$ subspace of codimension $2$ such that every $Q\_i$ must contain at least one of them. Now, it is not difficult to see that rank of $\{Q\_1, \ldots, Q\_k\}$ is bounded by $4$. The similar argument works for the rank of $\{P\_1, \ldots, P\_m\}$. 


