I conjecture the following:

Let $U \subset \Bbb C^{n \times n}$ be an affine subspace, and let $S_U$ denote the "spectrum of $U$", that is $$ S_U = \{\lambda \in \Bbb C : \det(A - \lambda I) = 0 \text{ for some } A \in U\}. $$ Then either all elements of $U$ have an identical spectrum, or $S_U = \Bbb C$.

Is this correct? Some simple examples of each case: for any fixed $\lambda_i$, $$ U_1 = \left\{\pmatrix{\lambda_1&t\\0&\lambda_2}: t \in \Bbb C\right\}, \quad U_2 = \left\{\pmatrix{\lambda_1&0\\0&t}: t \in \Bbb C\right\}. $$ Clearly, $S_{U_1} = \{\lambda_1,\lambda_2\}$ and $S_{U_2} = \Bbb C$. Are there any other possibilities? I suspect that there is a quick algebraic-geometry approach here that I am missing.

An aside: I would also be interested in the case of real affine subspaces of $\Bbb C^{n \times n}$, if anyone has insights on what the possibilities are there. Note that the symmetric real matrices are an example of a subspace where we attain $S_U^{(\Bbb R)} = \Bbb R \subsetneq \Bbb C$. We can clearly attain any "line" in $\Bbb C$, but I wonder if there are other possibilities.