# 2024-09-05-12-47-08

:::{.remark}
Last time: $S' \mapsvia{f} S$ with $S$ smooth and $f\inv$ undefined at $p\in S$.
Then $f$ contracts a nonempty curve $C \subseteq S'$, i.e. $f(C) = p$.
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:::{.remark}
Recall Zariski's main theorem: if $f:X\to Y$ is finite and birational with $Y$ normal, then $f$ is an isomorphism.
Recall that $Y$ is normal iff $\OO_{Y, p}$ is integrally closed in its fraction field, and smooth implies normal because UFDs are integrally closed.
We give two proofs of the theorem from last time.
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:::{.proof}
We will show that $f$ is finite and affine, i.e. preimages of affines are affines.
Recall a morphism $\spec A\to \spec B$ is finite if $B\to A$ makes $A$ a finite $B$-module.
Finite is equivalent to quasifinite (finite fibers) and proper.
Note that projective morphisms are proper, and any morphism between projective varieties is automatically projective.
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:::{.proof}
Recall $S$ is smooth, so $\OO_{S, p}$ is a UFD.
Embed $S' \subseteq \PP^n$ and consider $f\inv: U\to \AA^n \subseteq \PP^n$ where $U$ is an affine neighborhood of $p$.
Then $f\inv = ( \varphi_1, \cdots, \varphi_n)$ is given by $n$ rational functions.
Write $u/v$ for where $\phi_1$ is undefined at $p$, then $u(p) = v(p) = 0$.
Let $u/v$ be coprime.
Pullback and consider $f^*u, f^* v$.
Then $f^*u = f^*v \cdot x_1$ where $x_i$ are coordinates on $U$.
They are coprime in the UFD $\OO_{S, p}$.
So these differ by a regular function.
Consider the curve where $f^* v = 0$, then $f^* u = 0$.
By the principal ideal theorem, $f^*v=0$ is codimension one, hence a curve.
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:::{.remark}
Slogan: non-regular rational maps insert curves.
Note that it may factor as a sequence of blowups.
Recall Castelnuovo's contractibility criterion: rational $(-1)$-curves can be blown down.
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:::{.lemma}
If $S$ is smooth projective and contains $E\cong \PP^1$ with $E^2 = -1$, then $\exists f: S\to S'$ with $S'$ smooth and $S = \Bl_p S'$, i.e. $f(E) = p$ is a point and $\ro{f}{S\sm E}$ is an isomorphism.
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:::{.proof}
Consider $\OO_S(1) = \OO_{\PP^m}(1)\mid_S$ and let $H$ be a hyperplane.
Then $H$ is very ample.
Let $A = mH$ for $m\gg 0$; note $A$ is also very ample.
I.e. there is some $S\injects \PP^k$ where $\OO_S(m) = \OO_{\PP^k}(1)\mid_S$, this follows from pulling back a Veronese embedding $V_m: \PP^m\to \PP^k$.
The degree of the curve under this embedding is $A.E > 0$.
Define $L = A +mE$, then $L.E = 0$.
Claim: $\abs{L}$ is basepoint-free, contracts $E$ to a point, and the induced morphism $\phi_{\abs E}$ is an isomorphism outside of $E$.

Step one: $L$ is basepoint-free.
Consider $H^0(L)$.
We know $\Base(L) \subseteq E$ since the only possible zeros are along $E$.
Consider $H^0(L\mid_E)$ induced by the restriction $\OO(L) \surjects \OO_E(L)$.
Note that $\deg \OO_E(L) = 0$, and since $\Pic(\PP^n) = \ZZ$ we have $\OO_E(L) \cong \OO_{\PP^n}$.
There is a SES $0\to \OO_S(L-E) \to \OO_S(L) \to \OO_E(L)$, so by taking the LES, it suffices to show $H^1(\OO_S(L-E)) = 0$.
Now apply Serre's vanishing theorem: if $X$ is projective and $F\in \Coh(X)$, the twist $F(m)$ for $m\gg 0$ has vanishing higher cohomology and $F(m)$ is globally generated.
So $H^1(\OO_S(A)) = 0$, and $L = A + mE$, so $L-E = A + (m-1)E$.
We want $H^0(A + (m-1)E) = 0$.
There are SESs $0\to \OO_S(A + (k-1)E)\to \OO_S(A + kE)\to \OO_E(A+kE)$.
By induction, it suffices to show $H^1(\PP^1, \OO(m+k)) = 0$.
This is dual to $H^0(\PP^1, \OO(-2-m-k))$, which is zero if $k\leq m-1$.

> Missed details here, the markers were dying!

Delicate part of the argument: showing $S'$ is smooth.
It suffices to see that $\dim_\CC \mfm_p/\mfm_p^2 = 2$.
For that, we produce two explicit generators.
Take a SES $0\to \OO_S(L-2E)\to \OO_S(L-E)\to \OO_E(L-E) = \OO_{\PP^1}(1) \to 0$.
Take the LES, note $H^0(\OO_{\PP^1}(1)) = \gens{x,y}$ is 2-dimensional.
This is isomorphic to $H^0(L-E)/H^0(L-2E)$, we want to show this is $F\tensor \mfm_p/\mfm_p^2$ for $F$ some invertible sheaf.

Write $S' \injects \PP^n$ and $g: S\to \PP^n$ for the composition with $f$.
Then $L = g^* \OO_{\PP^n}(1)$.
Check that sections of $L-E$ vanish along $E$ to order at least 1, so are in $\mfm_p$.
Similarly sections of $L-2E$ vanish along $E$ to order at least 2 and are thus in $\mfm_p^2$.
Thus $F = \OO_{S'}(1)$.

Note $f_* \OO_S = \OO_{S'}$ and $L = f^* \OO_{S'}(1)$, so by the projection formula,
\[
f_* L = f_*\qty{f^*\OO_{S'}(1) } = f_* \OO_S \tensor \OO_{S'}(1) = \OO_{S'}(1)
.\]

Apply $f_*$ to $\OO(L-E)\injects \OO(L) \surjects \OO_E(L)$ and show one gets $\mfm_p(1) \to \OO_{S'}(1) \to \CC_p \to \RR^1 f_* \OO_S(L-E) = 0$.
Something similar must be shown for $L-2E$, but this requires a spectral sequence argument.
We instead appeal to Grothendieck's theorem on formal functions.
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:::{.remark}
Normalization produces a Stein factorization:

\[\begin{tikzcd}
	& {\tilde S'} \\
	\\
	S && {S'} & {\PP^n}
	\arrow[from=1-2, to=3-3]
	\arrow[from=3-1, to=1-2]
	\arrow[from=3-1, to=3-3]
	\arrow[hook, from=3-3, to=3-4]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsNCxbMCwyLCJTIl0sWzIsMiwiUyciXSxbMywyLCJcXFBQXm4iXSxbMSwwLCJcXHRpbGRlIFMnIl0sWzAsM10sWzMsMV0sWzAsMV0sWzEsMiwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=)

Here $\tilde S'$ is the normalization, and can be written as $\tilde S' = \spec f_* \OO_S$.

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:::{.remark}
Grothendieck's theorem on formal functions: let $f:X\to Y$ be projective, $F\in \Coh(X)$, and $p\in Y$.
Consider the higher direct images $\RR^i f_* F$ on $Y$, which is an $\OO_Y$-module.
Complete the stalk at $p$ to get $\widehat{\RR^i f_* F}_p$ which is a module over $\widehat{\OO_{Y, p}}$.
This can be computed as a limit $\colim_{Z} F\mid_Z$ where $Z$ are thickenings of the fiber $f\inv(p)$.
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:::{.remark}
We first apply this for $i=0$; we get $\widehat{f_* \OO_S} =\widehat{\OO_{S'}}$.
For smoothness, we want to show the latter is isomorphic to $\CC\fps{x,y}$.
The theorem says to compute $\colim_m H^0(mE)$.
We have $0\to \OO_S(mE)\to \OO_S \to \OO_{mE}$.
The claim is that the colimit is $\CC[x,y,x^2,xy,y^2,\cdots,y^m]$, the monomials up to the $m$th order.
Use the fact that $\OO_E(-mE) = \OO_{\PP^1}(m)$.
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:::{.remark}
This concludes chapter 2.
Next time: ruled surfaces.
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