Let $K$ be a field and let $A$ be a $K$-algebra. Suppose $A$ is contained in an affine $K$-domain $B$ (that is a finitely generated $K$-algebra that is an affine domain) and let $P \in \operatorname{Spec}(A)$.
My question is: Is $A/P$ also contained in an affine $K$-domain?
So far I have got that if $B$ is integral over $A$, then by lying over there is a $Q \in \operatorname{Spec} B$ s.t. $A \cap Q = P$, so there is a canonical injection $A/P \hookrightarrow B/Q$, which yields the desired property. This means that w. l. o. g. one can assume that $A$ is integrally closed over $B$.