Let $A \subseteq B$ be a finite ring extension of finitely generated $\mathbb{C}$-algebras of dimension one. Let $A$ be an integral domain. Consider the set $S$ of all maximal ideals $\mathfrak{m}\subseteq A$ such that $B/\mathfrak{m}B$ is reduced. Is it always true that $S$ is either empty or infinite?
The situation is clear to me when $B$ is reduced. In that case the set $S$ is infinite. Now if $B$ is not reduced, it can still happen that $S$ is infinite, but it may also be that $S$ is empty. But could $S$ also be just a nonempty finite set?