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:

Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements