Let $\mathcal C$ denote the commuting variety of pairs of matrices in $M_n(\mathbb{C})$, defined as:
$$ \mathcal C = \{ (A, B) \in M_n(\mathbb{C})^2 \mid [A, B] = 0 \}. $$
It is well known that $\mathcal C$ is irreducible of dimension $n^2 + n$.
Now, let $U$ be a vector subspace of $M_n(\mathbb{C})$. Consider the set of commuting pairs where the first matrix is restricted to $U$:
$$ \mathcal C_U = \mathcal C \cap (U \times M_n(\mathbb{C})). $$
Observations
- Certainly $\mathcal C_U$ contains the subset $\{ 0 \} \times M_n(\mathbb{C})$, so $\dim \mathcal C_U \geq n^2$.
- From computational experiments for small $n$ and generic $U$, I observed that:
- When $\mathrm{codim} \, U \leq n$, we have $\dim \mathcal C_U = \dim \mathcal C - \mathrm{codim} \, U$, so with each lower dimension $\dim \mathcal C_U$ decreases.
- For $\mathrm{codim} \, U \geq n$, the dimension of $\mathcal C_U$ stabilizes at $n^2$.
Questions
- Generality: Is it true that for generic choices of $U$, the dimension of $\mathcal C_U$ behaves as observed?
- Exceptions: Are there examples of particular subspaces $U$ where $\dim \mathcal C_U$ behaves unexpectedly?
- Stabilization: Do we always have $\dim \mathcal C_U = n^2$ for $\mathrm{codim} \, U = n$?
