$\textbf{1.}\,\,\,\,\,\,\,\,$ Let $(A,\mathfrak m_A)$ be a one dimensional local domain and let $B$ be its integral closure in the fraction field $L=\textrm{Frac}\,A$. Assume that $B$ is finitely generated over $A$. I would like to understand why the discrete valuations rings of $L$ dominating $A$ $are$ the localizations $B_{\mathfrak m}$ of $B$ at its maximal ideals.
I was able to show that such a localization is a DVR of $L$. But, just to conclude this first step, how does one prove that $\mathfrak m_A\subset \mathfrak mB_{\mathfrak m}$? We have \begin{equation} A\hookrightarrow B\hookrightarrow B_{\mathfrak m} \,\,\,\,\,\,\,\,\,x\mapsto x/1 \end{equation} but how do we know that $x\in\mathfrak m_A$ goes to something in $\mathfrak m$ via the first map? And, for showing the "converse" (from DVR to localization) I have no ideas.
$\textbf{2.}\,\,\,\,\,\,\,\,$ Afterwards, I would like to understand the following: suppose we have an integral variety $X$ with normalization $\pi:\tilde X \to X$. Let us take a (closed) codimension one subvariety $V\subset X$. The questions is: why do we have a correspondence between the DVR's of $L=K(X)$ dominating $A=\mathscr O_{X,V}$ and the (closed) subvarieties $Z\subset\tilde X$ mapping onto $V$? I feel like domination should translate $\pi(Z)=V$, but can't see it neatly.
Thank you!