# 2024-08-20-12-47-17: Algebraic Tools

## Divisors and $\Pic$

:::{.remark}
The tools we'll use: divisors, line bundles, sheaves, cohomology, intersection theory, and Riemann-Roch.
Almost all surfaces we consider in this class will be smooth.
Let $X$ be a smooth algebraic variety.
A divisor $D = \sum n_i Z_i$ with $n_i\in \ZZ$ and $Z_i$ codimension 1 subvarieties.
On a curve, divisors are weighted sums of points, and on a surface they are weighted sums of curves.
We say $D_1\sim D_2$ if $D_1 - D_2 = (\phi)$ with $\phi\in \CC(X)$ a rational function.
These are principal divisors.
We define $\Pic(X) \da \Div(X)/\sim$.

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:::{.example}
Fact: $\Pic(\PP^n) = \ZZ$.
Write $\PP^n = \ts{[x_0:\cdots:x_n]}/x\sim\lambda x$ and $H_i \da \ts{x_i\neq 0}$.
We have $\PP^n\setminus H_0 = \ts{[1:x_1:\cdots:x_n]}$.
The claim is that for any $D$, $D\sim dH_0$ for some $d$.
For $Z \subseteq \PP^n$ a codimension 1 subvariety, we have $\ro{Z}{H_0} = V(f)$ since $\CC[x_1,\cdots, x_n]$ is a PID.
Write $\phi = f_d\qty{ {x_1\over x_0}, \cdots, {x_n\over x_0}} = {g(x) \over x_0^d}$, which has a pole of order $d$ at $x_0$.
Thus $(\phi) = Z - dH_0$, so $Z\sim dH_0$.
Moreover, $dH_0 \sim 0 \iff d=0$.
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:::{.remark}
Over $\CC$, one has the GAGA principle: if $X \subseteq \CP^n$ is a closed analytic subset, by the Chow lemma it is algebraic.
One can then compute cohomology using analytic tools or algebraic tools.
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:::{.remark}
For example, one can identify $\Pic(X)$ as line bundles modulo isomorphism, i.e. $H^1(\OO_X\units)$.
Note that $\OO_X(U)$ are regular functions, and $\OO_X\units(U)$ are nowhere vanishing regular functions -- these are the first examples of sheaves.
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## Sheaves

:::{.remark}
Recall that a sheaf $F$ of abelian groups is an assignment of open sets $U$ to sections $F(U)$ with appropriate restriction maps.

\includesvg{inkscape/2024-08-20_13-05.svg}



Recall that $\OO_X(D)$ is locally isomorphic to $\OO_X$, and sections of $\OO_X(D)$ are rational functions $\psi$ such that $(\psi) + D \geq 0$.
Recall that $D_1\sim D_2 \implies \OO_X(D_1)\cong \OO_X(D_2)$ by writing $D_1-D_2 = (\zeta)$ and mapping $s\mapsto s\zeta$.
Checking $(\zeta) + D_1\geq 0$, one has 
\[
(\zeta \psi) + D_2 \geq 0 \iff (\zeta) + (\psi) + D_2 \geq 0 \iff D_1-D_2 + (\psi) + D_2 \geq 0 \iff D_1 + (\psi) + D_1 \geq 0
.\]

We say $\OO_X(D)$ is locally free of rank 1, or invertible, or a line bundle.

\includesvg{inkscape/2024-08-20_13-16.svg}

Note that $\Aut(\OO_X) = \OO_X\units$.
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:::{.remark}
There is a SES 
\[
0\to I\to \OO_X \to \OO_C \to 0
\]
where $I\cong \OO_X(-C)$.
These are examples of coherent sheaves, those which can locally be written $\OO^{\oplus J}\to \OO^{\oplus I}\to F\to 0$.
An example of a non-coherent sheaf is $\OO_X\units$, since one can not multiply a section by a function with a zero.
Other examples include locally constant sheaves $\ul{\ZZ}, \ul{\CC}$.
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:::{.remark}
Given $A\injects B\surjects C$ there is a LES $H^i(A)\to H^{i}(B)\to H^{i}(C)\to H^{i+1}(A)\to\cdots$.
By Grothendieck, $H^i(F) = 0$ for $i > d = \dim X$ if $F$ is coherent and $X$ is a Noetherian topological space.
If $X$ is projective and $F$ is coherent, $H^i(F)$ all form finite-dimensional vector spaces.
We write these dimensions as $h^i(F)$, which are numerical invariants.
Define $\chi(F) = \sum_i (-1)^i h^i(F)$.
One has $\chi(B) = \chi(A) + \chi(C)$, since $\chi$ of a LES is zero.
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:::{.remark}
Recall that the Betti numbers are defined as $\beta_i \da \dim_\QQ H^i(X;\QQ)$ and $\chi(X) \da \sum (-1)^i \beta_i$.
Note that $\chi(X) \neq \chi(\OO_X)$ in general: let $C$ be a smooth projective curve of genus $g$ over $\CC$.
Check that $\chi(X) = 1 - 2g + 1 = 2-2g$, since there are $2g$ generators for homology:

\includesvg{inkscape/2024-08-20_13-41.svg}

However, recall $h^1 = h^{1, 0} + h^{0, 1} = g+g$, and $\chi(\OO_X)= h^0(\OO_X) - h^1(\OO_X) = 1-g = {1\over 2}(2-2g)$ instead.
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## Cohomology

:::{.remark}
Recall the exponential exact sequence $0\to \ZZ\mapsvia{\cdot 2\pi i} \OO_X \mapsvia{\exp} \OO_X\units\to 1$.
Taking the LES yields:

\[\begin{tikzcd}
	{H^0(\ZZ) = \ZZ} && {H^0(\OO_X) = \CC} && {H^0(\OO_X\units) = \CC\units} \\
	\\
	{H^1(\ZZ) = \ZZ^{2g} \oplus \mathrm{tors}} && {H^1(\OO_X) = \CC^g} && {H^1(\OO_X\units) = \Pic(X)} \\
	\\
	{H^2(\ZZ) = \ZZ^n\oplus\mathrm{tors}} && {H^2(\OO_X)} && \cdots
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-5, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-5, to=5-1]
	\arrow[from=5-1, to=5-3]
	\arrow[from=5-3, to=5-5]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsOSxbMCwwLCJIXjAoXFxaWikgPSBcXFpaIl0sWzIsMCwiSF4wKFxcT09fWCkgPSBcXENDIl0sWzQsMCwiSF4wKFxcT09fWFxcdW5pdHMpID0gXFxDQ1xcdW5pdHMiXSxbMCwyLCJIXjEoXFxaWikgPSBcXFpaXnsyZ30gXFxvcGx1cyBcXG1hdGhybXt0b3JzfSJdLFsyLDIsIkheMShcXE9PX1gpID0gXFxDQ15nIl0sWzQsMiwiSF4xKFxcT09fWFxcdW5pdHMpID0gXFxQaWMoWCkiXSxbMCw0LCJIXjIoXFxaWikgPSBcXFpaXm5cXG9wbHVzXFxtYXRocm17dG9yc30iXSxbMiw0LCJIXjIoXFxPT19YKSJdLFs0LDQsIlxcY2RvdHMiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XSxbNCw1XSxbNSw2XSxbNiw3XSxbNyw4XV0=)

We thus get a SES $0\to \CC^g/\ZZ^{2g} \to \Pic(X) \to F \to 0$ where $F$ is some finitely generated abelian group.
We define $\Pic^0(X) = \Jac(X) = \CC^g/\ZZ^{2g}$, which is an algebraic torus over $\CC$ of dimension $g$, and is the continuous part.
We identify $F \da \NS(X)$ as the Néron-Severi group.
Note that divisors in $\Pic^0(X)$ are numerically zero, since intersection numbers are integers that vary continuously.
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:::{.example}
$\NS(\PP^n) = \ZZ$, and $\NS(C) = \ZZ$ for a genus $g$ curve since the only numerical invariant of continuously varying divisors is $\sum n_i$.
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:::{.remark}
Note that $D_1, D_2\in \Pic^0(X)$ means $D_1$ is algebraically equivalent to $D_2$, i.e. can be varied continuously.
For linear equivalence, one uses $\PP^1$ as the base, since rational functions on $X$ are equivalent to maps to $\PP^1$.
Algebraic equivalence allows for the base to be an arbitrary curve.
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