Is $A/\varphi^{-1}(\mathfrak{m})\subseteq B/\mathfrak{m}$ an integral extension?
If $B$ is finitely generated as a $k$-algebra, and $\phi:A\to B$ is a $k$-algebra map, is $\phi^{-1}(M)$ maximal for any maximal $M\subset B$?