Let $\mathbb C[x_1,\ldots,x_n]$ be the polynomial ring of $n$ varibles, and $\mathbb C[x_1,\ldots,x_n]_d$ be its homogeneous degree $d$ piece. Let $F\in \mathbb C[x_1,\ldots,x_n]_3$ be a homogeneous cubic polynomial which has smooth zero locus. Then $$P=\mathbb C\{x_i \partial_j F\}$$ is a linear space of dimension $n^2$ (by smoothness). For any homogeneous quadratic polynomial $q \in \mathbb C[x_1,\ldots,x_n]_2$, we define $$n(q)=dim\{f\in \mathbb C[x_1,\ldots,x_n]_1: f\cdot q \in P \}$$ which can be also formulated as the dimension of the set of linear polynomial $f$ whose multiplication with $q$ vanishes in the Jacobi ring.
For example, easy to see $n(q)=n$ if and only if $q=\sum c_i \partial_i F$.
I computed several examples, and it seems the following is always hold:
If $q\neq \sum c_i \partial_i F$, then $n(q)\leq2$.
But I don't know how to prove it. I can show it in some special case, but in general $\{\partial_iF\}$ is quite mysterious to me. Also it seems the function $n(q)$ is hard to control.
Could someone help me? Thanks in advance!