Assume $X$ is an reducible affine variety over $\mathbb{C}$ with coordinate ring $A$. Let $\pi \colon \tilde{X} \to X$ be the blowing up at a point $p \in X$ (which can be included in more than one component of $X$). I have a hard time with the following problems:
If $\dim X = 1$, then the fiber $\pi^{-1}(p)$ should be finite. How can I prove this elegantly
If the fiber $\pi^{-1}(p)$ is finite ($\dim X$ arbitrary) $\tilde{X}$ would be affine, because I can find a chart which includes all the points in the fiber, is this a correct argumentation?
Assume $\tilde{X}$ affine with coordinate ring $B$. I have a ring homomorphism $\phi \colon A \to B$ induced by $\pi$. Under which conditions can I assume, that $\phi$ is injective?
Assume $\tilde{X}$ affine with coordinate ring $B$. Is the total ring of fractions isomorphic to the total ring of fractions of $A$? How can I show this? This would be clear, for $X$ irreducible, because $\tilde{X}$ is birational to $X$.
I have seen most of the questions answered in literature for irreducible varieties (or integral schemes), but have problems to see this, if $X$ is not irreducible, i.e. $A$ not integral. Any help for any of the points 1-4 would be greatly appreciated