Let $k$ be an infinite perfect field (e.g. I'm happy to assume that $k$ has characteristic $0$. On the other hand, the algebraically closed case is not interesting for this question). The question is happening inside the vector space $k^{21}$.
Let $f$ be a homogeneous polynomial of degree 6 in 21 variables with coefficients in $k$. For $\lambda \in k$, let $S_{f=\lambda}$ be the corresponding level set of $f$ in $k^{21}$, i.e. $S_{f=\lambda}:=\lbrace x\in k^{21}~\vert~f(x)=\lambda \rbrace$.
QUESTION: What kind of conditions can one give on $f$ to ensure that for any $8$-dimensional vector subspace $W<k^{21}$, the set $W\cap \big( \bigcup \limits_{\lambda \in (k^{\times})^2} S_{f=\lambda} \big)$ is non-empty?
I hope that the given parameters $21,6$ and $8$ are not really relevant, but this is what I get in my specific situation. Any comment on how to think about this question is welcome!
