Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be  an irreducible homogenous polynomial.

Is there a  geometric or algebra geometric interpretation for the following quantity:

The maximum number $k$ such that there is  a  $k$  dimensional subvector  space $Y$ of $\mathbb{C}^{n}$  which is  included in $P=0$?

 Moreover,  what is this  quantity for $Det:M_{n}(\mathbb{C})\simeq \mathbb{C}^{n^{2}}\to \mathbb{C}$?

This  question is motivated by the following post:



https://mathoverflow.net/questions/195950/finite-codimensional-subvector-space-of-c-algebras-which-contains-no-inver?noredirect=1#comment488733_195950