My questions refer to some arguments used in Liu's "Algebraic Geometry and Arithmetic Curves" in Chapter 8.1.4 (p 330). Here the excerpt:
We consider the blow-up (I'm working with the definition given at page 318) $\pi_n: X_n \to X_{n-1}$. from prop. 1.12 (page 322) we know that $\pi_n$ is bitational and proper morphism and $X_n, X_{n-1}$ are integral curves (so $1$-dimensional proper, $k$-schemes).
My questions are:
how the author deduces that then $\pi_n$ is a finite morphism? Especially why affine?
In 1.26 is shown that if $X$ is integral projective curve then the blow up sequence is finite. I don't understand the argument given at remark 1.27 that says that this also holds without the assumtion $X$ is projective. How does the argument works?
Below I attatched the definition of a blowing up used by Liu:
Let's write our blowup morphism as $B:\widetilde{X}\to X$, where we assume that $\widetilde{X},X$ are integral 1-dimensional schemes over $k$. Note that $B$ is birational and proper. We proceed in two steps: first we show that $B$ is quasifinite, then we conclude that $B$ is in fact finite.
We know that over the dense subset $U\subset X$ where $B$ is an isomorphism, the fiber over any point is a single point, so it remains to show that $B$ is quasifinite outside of $U$. As properness is invariant under base change, so we know that the fiber over every point in $X$ is proper. Now I claim that for any $x$ not in $U$, the fiber over $x$ cannot be positive-dimensional, as otherwise $\widetilde{X}$ would not be integral. Why? Suppose $B^{-1}(X)$ is positive dimensional. Then $\widetilde{X}$ would not be locally irreducible at any point in the strict transform $\overline{B^{-1}(U)}$ which also lies in the fiber over $x$, but every integral scheme is locally irreducible at every point. As $0$-dimensional proper $k$-schemes have finitely many points, $B$ is quasifinite.
By quasifinite plus proper implies finite, we know that $B$ is finite. Since finite is equivalent to affine plus proper, this shows that $B$ is also affine.
You'll notice that at no point did we every use the assumption that $X$ was proper or projective, just integrality of $\widetilde{X}$ and properness/birationality of $B$. So there's no reason to require $X$ be projective.