# 2024-08-27-12-46-20: Intersection theory on surfaces

:::{.remark}
Recall that $\NS(X)$ carries an integral intersection form.
For curves without common components, it counts the intersection points with multiplicities.

\includesvg{inkscape/2024-08-27_12-47.svg}

For all $f:X\to Y$ maps of varieties, define $f^*: \Pic(Y)\to \Pic(X)$ and $\NS(Y)\to \NS(X)$, defined by pulling back equations.
Define $f_*: \Pic(X)\to \Pic(Y)$ by
\[
f_* C = 
\begin{cases}
0 & f(C) = \pt \\
dC' & f(C) = C', d=\deg(C\to C')
\end{cases}
.\]

Some properties:

1. If $f: S'\to S$ is generically degree $d$ then 
\[
(f^* D_1)\cdot (f^* D_2) = d D_1 \cdot D_2
.\]

2. Projection formula:
\[
f^* D_1 \cdot D_2 = D_1 \cdot f_* D_2
.\]


3. Hirzebruch-Riemann-Roch: let $X$ be a smooth projective of dimension $n$ and $L = \OO_X(D)$ a line bundle on $X$.
Then
\[
\chi(L) = \int_X e^D \mathrm{Td}_X
.\]
Here $e^D = \sum_k D^k/k!$ and $\mathrm{Td}_X = 1 + {1\over 2}c_1 + {1\over 12}(c_1^2 + c_2) + \cdots$ and $c_i$ are the Chern numbers of $T_X$.


Recall that $c_1: \Pic(X)\to H^2(X; \ZZ)$.
Note that $c_1 = -K_X$ and $c_n = \chi_{\Top}(X)$.
Here $\int_X(\wait)$ is the degree of a cycle of codimension $n$, hence dimension 0, and counts its number of points.
:::

:::{.example}
Let $n=1$, so $C$ is a curve of genus $g$.
Then $c_1 = -K_C$ has degree $2-2g$, and $c_n = c_1 = \chi_\Top(C) 1 - 2g + 1 = 2-2g$.
Thus HRR yields 
\[
\chi(D) = \int_X (1+D)(1 + {1\over 2}c_1) = \deg\qty{D + {c_1\over 2}} = \deg D + 1 - g
.\]
Setting $D=0$ yields $\chi(\OO_C) = \deg\qty{c_1\over 2} = 1-g$.
:::

:::{.example}
\[
\chi(\OO_S(D)) = \int_S \qty{1 + D + {1\over 2}D^2} \qty{1 + {1\over 2}c_1 + {1\over 12}(c_1^2 + c_2)} = {1\over 2}D^2 + {1\over 2}Dc_1 + {1\over 12}(c_1^2 + c_2)
,\]
thus $\chi(\OO_S) = {1\over 12}(c_1^2 + c_2)$ by setting $D=0$, which recovers Noether's formula.
Note that $c_1 = -K_S$, so $\chi(\OO_S) = {K^2 + c^2 \over 12}$.
We can then rewrite this as
\[
\chi(D) = {D(D-K)\over 2} + \chi(\OO_S)
\]
where $\chi(\OO_S) = 1-q+p_g$.
:::

:::{.remark}
Recall that for a curve $D$ on a surface $S$, one has $\deg K_D = 2p_a(D) - 2$ and $K_D = \qty{K_S + D}\mid_D$.
From the genus formula, $p_a(D) = {D(D+K)\over 2} + 1$.
:::

:::{.example}
Consider $S = \PP^2$, so $K_S = -3H$ and $K_S^2 = 9$.
Check that $\chi_\Top(S) = 3$ by counting torus fixed points in the standard polytope, and thus ${c_1^2 + c_2 \over 12} = {9+3\over 12} = 1$.
Check that
\[
h^0(\OO_{\PP^2}(d)) = 
\begin{cases}
{d+2\choose 2} & i=0 \\
h^0(-3-d) & i=2 \\
0 & i\neq 0, 2
\end{cases}
.\]
Thus $\chi(\OO_S) = 1$.
:::

:::{.example}
Let $S =\PP^1\times \PP^1$, so $K = \OO(-2, -2) = 2e-2f$ and $K^2 = 8$ since $ef=1$.
Moreover $\deg c_2 = 4$ by considering the toric polytope and counting points.
Noether's formula yields ${8+4\over 12} = 1$.
:::

:::{.remark}
Note that $\chi_\Top(\OO_S(D))$ can be computed easily for toric varieties.
The Euler characteristic is additive, and every such variety decomposes as $(\CC\units)^2 \disjoint \CC\units \disjoint \ts{p_i}$, and $\chi(\CC\units) = 0$, so only points $p_i$ contribute.
:::

## Birational geometry of surfaces

:::{.remark}
Recall that rational maps on irreducible varieties are regular functions on open subsets, and correspond to maps $X\to \AA^1$.
Note that any two open subsets of an irreducible variety intersect.
One can add, multiply, and invert nonzero rational functions by throwing out closed sets of zeros, so these form a field.
Say $X\birational Y$ are birationally isomorphic if they share a common open set $X \contains U \subseteq V$.
Say $X$ is rational iff $X\birational \PP^n$ or $\AA^n$, and $X$ is unirational if there exists a dominant rational map $\PP^n\rational X$ (covered by a rational variety).
Dominant morphisms: closure of the image is the entire space.
Note that dominant rational maps $X\to Y$ yields an embedding of fields $\CC(Y) \injects \CC(X)$.
Thus rational varieties of dimension $n$ satisfy $\CC(X) = \CC(x_1,\cdots, x_n)$, while unirational means $\CC(X)$ is a subfield.
:::

:::{.remark}
Luroth problem: does unirational imply rational?
True for $n=1,2$, we will prove that $n=2$ case in this course.
Generally false for $n\geq 3$.
:::

:::{.theorem}
Suppose that $S_1\birational S_2$ are birationally isomorphic smooth projective surfaces.
Then there exists a diagram of blowups and blowdowns forming a correspondence:

\[\begin{tikzcd}
	&&& S \\
	&& \bullet && \bullet \\
	& \bullet &&&& \bullet \\
	{S_1} &&&&&& {S_2}
	\arrow[from=1-4, to=2-3]
	\arrow[from=1-4, to=2-5]
	\arrow["{\text{blowups}}"', from=2-3, to=3-2]
	\arrow["{\text{blowdowns}}", from=2-5, to=3-6]
	\arrow[from=3-2, to=4-1]
	\arrow[from=3-6, to=4-7]
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsNyxbMywwLCJTIl0sWzIsMSwiXFxidWxsZXQiXSxbMSwyLCJcXGJ1bGxldCJdLFswLDMsIlNfMSJdLFs0LDEsIlxcYnVsbGV0Il0sWzUsMiwiXFxidWxsZXQiXSxbNiwzLCJTXzIiXSxbMCwxXSxbMSwyLCJcXHRleHR7Ymxvd3Vwc30iLDJdLFsyLDNdLFswLDRdLFs0LDUsIlxcdGV4dHtibG93ZG93bnN9Il0sWzUsNl1d)

:::

:::{.remark}
We will define a blowup of a smooth surface at a smooth point $\pi: \Bl_p S\to S$.
We start with the example of $S=\AA^2$ and $p$ an arbitrary point.
This will be birational because $\AA^2\setminus\ts{p} \cong \Bl_p\AA^2\setminus E$.
Different lines passing through $p$ will become disjoint curves intersecting $E$.

\includesvg{inkscape/2024-08-27_13-42.svg}

In equations, write $\Bl_p \AA^2 = \ts{xY=yX} \subseteq \AA^2 \times \PP^1$ with coordinates $x,y$ and $X,Y$ respectively.
Note that $x/y = X/Y$ if $y,Y\neq 0$.
:::

:::{.remark}
In general, for $\tilde S\to S$, let $x,y$ be local parameters for $p\in S$.
One takes $\OO_{S, p}\contains \mfm_p = \gens{x,y}$ with $T_p\dual X = \gens{dx, dy}$ and $\hat{\OO_{S, p}} = \CC\fps{x,y}$.
So any local regular function $s$ can be approximated by a power series.
:::

:::{.remark}
Claim: $E^2 = -1$.
Write $\PP^1 = \AA^1\union \AA^1$ with $u=X/Y$ and $v=Y/X$.
Write $\Bl_p \AA^2 = \AA^2_{u, y} \union \AA^2_{v, x}$.
This is a change of coordinates 

- $u,y = {x\over y}, y$ where $x=uy$, 
- $v,x = {y\over x}, x$ where $y=vx$.

If $C$ is a line through $p\in \AA^2$, which can be replaced with a curve $C$ that is smooth at $p$, then $f^*C = C' + E$ where $C' = \overline{f\inv(C \setminus p)}$ is the strict transform.
Write $C = \ts{g=0} = \ts{y+\alpha x = 0}$.
Then $f^* g = vx + ax = x(v+a)$ where $x=0$ is the equation of $E$ and $v+a$ is the equation of $C'$.
Conclude using the projection formula: $f^* C \cdot E = C \cdot f_* E = C\cdot 0$ on one hand, and is equal to $(C'+E)E = C'E + E^2 = 1+E^2$ on the other hand, so $E^2 = -1$.
:::