# 2024-08-22-12-46-47: Divisors and Intersection Theory

## Divisors

:::{.remark}
Let $X$ be a smooth projective variety, and recall $\Pic(X) = \Div(X)/\sim$ or the group of rank 1 locally free sheaves $F$ where $\ro{F}{U} = \ro{\OO_X}{U}$.
Both are isomorphic to $H^1(\OO_X\units) \cong \check{H}^1(\OO_X\units)$, the \v{C}ech cohomology.
Let $\mcu$ be an open cover of $X$, so $X = \union_i U_i$.
Then a divisor $D$ on $U_i$ locally has an equation $f_i$ for some rational function, i.e. $(f_i) = D$.
Consider $\ts{f_i \in K\units(U_i)}$.
Then on $U_{ij}$, write $g_{ij} = {f_i \over f_j}$.
This has no zeros and no poles, so $(g_{ij}) = 0$.
Thus $g_{ij}$ and $1/g_{ij}$ are both regular and give sections $g_{ij}\in \OO\units(U_{ij})$.
We get a collection $\ts{g_{ij} \in \OO\units(U_{ij}) \st g_{ij}\cdot g_{jk} \cdot g_{ki} = 1 \text{ on } U_{ijk}}$ modulo $f_i' = f_i g_i$ for $g_i\in \OO\units(U_i)$.
This is precisely $\check{C}^1/\check{B}^1 = \check{H}^1(\mcu, \OO_X\units)$.
One defines $\check{H^1}(X, F) = \colim_{\mcu} \check{H}^1(\mcu, F)$ for a sheaf $F$.
:::

:::{.example}
Let $F = \OO_{\PP^1}(H)$ where $H\da p_\infty \da \ts{x_0 = 0}$ in coordinates $[x_0: x_1]$.
Write $\PP^1 = \AA^1_{x_1\over x_0} \union \AA^1_{x_0\over x_1}$.
The equations of $D$ are $1$ and ${x_0\over x_1}$ respectively, and the transition function is $f_{12} = {x_0\over x_1}$.
Thus $F = \OO(1)$.

Letting $D = dH$ yields $\OO(D) = \OO(d)$.
Note that $\PP^n\setminus H \cong \AA^n$ where $H = \ts{x_0 = 0}$.
:::

:::{.remark}
For any projective variety, there are three important sheaves:

- $\OO_X$,
- $\ro{ \OO_{\PP^n}(1) }{X} = \OO_X(1)$ for $X \injects \PP^N$ a projective embedding,
- $\omega_X = \OO_X(K_X)$.

On a normal variety, $\CDiv(X) \injects \Div(X)$ is a subgroup.
There is always such a map, even for non-normal varieties, but generally $\CDiv(X)$ is bigger.
:::

:::{.remark}
Letting $\mcu$ be an open cover, consider $U_{ij}$.
We have $\ro{F}{U_i}\cong \OO_{U_i}$, and similarly we get a diagram

\[\begin{tikzcd}
	{\ro{F}{U_{ij}}} && {\OO_{U_{ij}}} & {g_{ij}\in \OO\units(U_{ij})} \\
	\\
	{\ro{F}{U_{ij}}} && {\OO_{U_{ij}}} & 1
	\arrow["{\alpha_i}"', hook, two heads, from=1-1, to=1-3]
	\arrow[Rightarrow, no head, from=1-1, to=3-1]
	\arrow["{\alpha_j}", hook, two heads, from=3-1, to=3-3]
	\arrow[from=3-3, to=1-3]
	\arrow[maps to, from=3-4, to=1-4]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=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)

:::

:::{.remark}
If $\dim X = n$, then $\omega_X \da \Omega_{X}^n$ which has sections $\Omega_X(U) = \ts{f dx_1 \wedge \cdots \wedge dx_n }$ where $f$ is regular.
This is an invertible rank 1 locally sheaf.
We have $\Omega_X^1 = \ts{f_1 dx_1 + \cdots + f_n dx_n}$ which is rank $n$, locally isomorphic to $\OO_X^{\oplus n}$ since the $dx_i$ form a basis.
Similarly we have $\rank \Omega_X^k = {n\choose k}$.
:::

:::{.example}
Consider again $\PP^1 = \AA^1_u \union \AA^1_v$,
where $u={x_0\over x_1}$ and $v={x_1\over x_0}$, so $v=1/u$.
On the intersection, $dv = d(1/u) = -{du\over u^2}$ and thus $\omega_{\PP^1} = \OO_{\PP^1}(-2)$ and $K_{\PP^1} = -2H$ where $H$ is a point.
More generally, $K_{\PP^n} = -(n+1)H$ and $\omega_{\PP^n} = \OO(-n-1)$.

Similarly, $K_{\PP^1\times \PP^1}= p_1^*(-2H) + p_2^*(-2H)$ where $p_i$ are the two projections.
More generally, $K_{X\times Y} = p_1^*K_X + p_2^* K_Y$.
Note that $\Pic(\PP^1\times \PP^1) = \ZZ\oplus \ZZ$, but $\Pic(X\times Y)\neq \Pic(X) \oplus \Pic(Y)$ in general.
For a counterexample, take $E$ an elliptic curve, then $\Pic(E\times E) \contains \ZZ^3$.
We write $\OO(a, b) \in \Pic(\PP^1, \PP^1)$ to denote bidegree $a,b$ curves.
:::

:::{.remark}
To compute for other varieties, we do this indirectly using the adjunction and Hurwitz formula.
For $Y \subseteq X$, we have $K_Y = (K_X + Y)\mid_Y$, or $\omega_Y = (\omega_X \tensor \OO_X(Y))\mid_Y$.
For $C \injects \PP^2$ a curve of degree $d$, this yields $K_C = (d-3)H\mid_C$.
One checks $2g-2 = \deg K_C = d(d-3)$, which implies $g = {1\over 2}(d-1)(d-2)$.
:::

:::{.exercise}
For $C\times C$, show that $\Delta^2 = 2-2g$.
On the other hand, show $(f_1 + f_2)^2 = 2$, so $\Delta\not\sim af_1 + bf_2$ for any fibers $f_i$.
:::

:::{.remark}
Consider $S_d \injects \PP^3$ a hypersurface of degree $d$ cut out by some $f_d$.
Then $K_S = (d-4)H\mid_S$, and $d=4\implies K_S = 0$ and this gives a K3 surface.
Similarly $d<4 \implies K_S > 0$ and $d<4\implies K_S < 0$.
:::

:::{.remark}
The Hurwitz formula: let $\pi: X\to Y$ be a map of curves, then $K_X = \pi^* K_Y + R$ where $R = \sum(n_i - 1)p_i$ where $n_i$ is the ramification number -- given in local equations as $y=x^{n_i}$.
Note that if $y=x^n$ then $dy = nx^{n-1} dx$.
This equation generalizes to ramified maps of surfaces with the same formula.

Generally for a 2-to-1 cover $\pi: S\to \PP^2$, one has $R = {1\over 2}\pi^*B$ and 
\[
K_S = \pi^* K_{\PP^2} + R = \pi^*(K_{\PP^2} + {1\over 2} B)= \pi^* (-3 + d/2)H
.\]
:::

:::{.remark}
Recall that $h^{i}(D) = h^{n-i}(K-D)$ by Serre duality.
Define 

- $q\da h^1(\OO)$ the irregularity,
- $p_g \da h^2(\OO)$ the geometric genus,
- $h^1(\omega) = h^1(\OO) = q$,
- $h^2(\omega) = h^0(\OO) = 1$,
- $h^0(\omega) = h^2(\OO) = p_g$,
- $h^0(mK) = p_m$ the plurigenera,
- Consider $\lim_{m\to \infty} p_m \sim m^{\kappa}$, then $\kappa$ is the Kodaira dimension.

:::

## Intersection Theory

:::{.remark}
There is a symmetric pairing $\NS(X)\times \NS(X)\to \ZZ$ given by intersecting curves and counting points.
Recall there was a SES
\[
0\to {H^1(\OO_X) \over H^1(X;\ZZ)}\to \Pic(X) \to \NS(X) \da \ker\qty{H^2(X;\ZZ)\to H^2(\OO_X)}\to 0
.\]
We have $\NS(X)/\tors \cong\ZZ^{\rho}$ where $\rho$ is the Picard rank.
Note that this is compatible with the cup product pairing when working with varieties over $\CC$.
A definition that works over any field: $C_1\cdot C_2 = \sum_{p\in C_1 \intersect C_2} \mu_p(C_1, C_2)$ where $\mu_p(C_1, C_2) = \dim_\CC \OO_{S, p}/\gens{f, g}$ where $f,g$ are local equations for the $C_i$ near $p$.
This works when $\size(C_1 \intersect C_2)< \infty$.
:::

:::{.remark}
The moving lemma: $\forall C_2 = A-B$ where $A, B$ are general hyperplanes for some $S \injects \PP^{n_i}$.
Note that if $C\injects S$ is smooth, then $C^2 = \deg N_{C/S}$, the degree of the normal bundle.
:::

Upcoming: Hirzebruch-Riemann-Roch.