# 2024-10-08-12-51-02

:::{.remark}
Last time: del Pezzo surfaces, where $-K_S$ is ample.
Note that for del Pezzos, $-K_S$ is also effective, and is of the form $S = \PP^1\times \PP^1$ or $S = \Bl_k \PP^2$ where $0\leq k \leq 8$.
:::

:::{.example}
Let $S = \Bl_8 \PP^2$, then there always exists a curve $C$ through these 8 points whose preimage satisfies $[-K_S] = [\tilde C] = 3H - \sum E_i$.
:::

:::{.remark}
Generally, $L$ is an effective line bundle iff $L$ has a nonzero section $s$.
By intersecting $s$ with the zero section of $L$, one obtains a divisor $D$.
In the 1-dimensional case, this is the correspondence $D\mapstofrom L\da \OO(D)$.
:::

:::{.remark}
Question: is $-K_X$ effective for any (smooth) Fano of any dimension over $\CC$?

- Dimension 2: true.
- Dimension 3: true (Sokurov, 1980).
- Dimension 4: true (Kawamata, 2000).
- Dimension $n\geq 5$: open.

:::

:::{.remark}
Generally ample does not imply effective.
For example, let $C$ be a genus 2 curve and $L$ a degree 1 line bundle.
Any positive degree line bundle on a curve is ample.
Consider the moduli space $\Pic^1(C)$ of degree 1 line bundles on $C$.
This is isomorphic to the Jacobian of $C$, an abelian variety of dimension 2.
If $L$ is effective as a line bundle, then $L\cong \OO(D)$ with $D$ effective.
By definition, $D$ is effective iff $D =\sum a_i D_i$ with $a_i > 0$.
Since $\deg L = 1$, without loss of generality we can write $a_1 = 1$ and thus $a_i = 0$ for $i\neq 1$.
So $L = \OO(a_1 p)$ where $p\in C$ is a point.
Thus the moduli space of such bundles is in bijection with the points of $C$, and is dimension 1.
Since $\Pic^1(C)$ includes ample line bundles, there are many ample but non-effective bundles.
:::

:::{.remark}
For any algebraic curve $C$, there is an Abe-Jacobi map given by $\phi: \Sym^d(C)\to \Pic^d(C)$.
Note that $\Sym^d(C) = C^d/S_d$ where $S_d$ is the symmetric group, and is the moduli space of degree $d$ effective divisors on $C$.
On the other hand, $\Pic^d(C)$ is the moduli space of degree $d$ line bundles.
This map is given by $D\mapsto \OO(D)$, and is not surjective in general.
:::

:::{.remark}
Recall that $X$ is rational if $X\birational \PP^n$.
In dimension 1, $\PP^1$ is the only smooth projective rational curve.
In dimension 2, any del Pezzo is rational.
Questions:

- Can we classify all rational surfaces?
- Can we find criteria to decide if a surface is rational or not?

:::

:::{.remark}
Recall that $\Omega^k_X$ is the vector bundle of algebraic $k$-forms on a smooth projective variety $X$ of dimension $n$.
These are used to define Hodge numbers $h^{k, 0}(X) = \dim H^0(\Omega^k_X)$, which are birational invariants.
:::

:::{.remark}
For $X = \PP^n$, one has $h^{k, 0} = 0$ for $k\neq 0$ and $h^{0, 0} = 1$.
For $\PP^2$, this yields the following Hodge diamond:

\[\begin{tikzcd}
	&& 1 \\
	& 0 && 0 \\
	0 && 1 && 0 \\
	& 0 && 0 \\
	&& 1
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsOSxbMiwwLCIxIl0sWzEsMSwiMCJdLFszLDEsIjAiXSxbMCwyLCIwIl0sWzIsMiwiMSJdLFs0LDIsIjAiXSxbMSwzLCIwIl0sWzMsMywiMCJdLFsyLDQsIjEiXV0=)

Thus any rational surface should have a Hodge diamond of the following form:

\[\begin{tikzcd}
	&& {h^{2,2}} \\
	& {h^{2,1}} && {h^{1,2}} \\
	0 && {h^{1,1}} && {h^{2,0}} \\
	& 0 && {h^{1,0}} \\
	&& 1
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsOSxbMiwwLCJoXnsyLDJ9Il0sWzEsMSwiaF57MiwxfSJdLFszLDEsImheezEsMn0iXSxbMCwyLCIwIl0sWzIsMiwiaF57MSwxfSJdLFs0LDIsImheezIsMH0iXSxbMSwzLCIwIl0sWzMsMywiaF57MSwwfSJdLFsyLDQsIjEiXV0=)

One can conclude, for example, that K3 surfaces are not rational.
:::

:::{.remark}
Question: is any surface $S$ with $h^{1, 0}(S) = h^{2,0}(S) = 0$ necessarily rational?
Answer: no, consider Enriques surfaces, which are not rational:

\[\begin{tikzcd}
	&& 1 \\
	& 0 && 0 \\
	0 && 10 && 0 \\
	& 0 && 0 \\
	&& 1
\end{tikzcd}\]
> [Link to Diagram](https://q.uiver.app/#q=WzAsOSxbMiwwLCIxIl0sWzEsMSwiMCJdLFszLDEsIjAiXSxbMCwyLCIwIl0sWzIsMiwiMTAiXSxbNCwyLCIwIl0sWzEsMywiMCJdLFszLDMsIjAiXSxbMiw0LCIxIl1d)

:::

:::{.remark}
We instead introduce more refined birational invariants in terms of $h^0( (\Omega^k)^{\otimes m} )$ for $m > 0$.
For $\PP^n$, these vanish for all $m$.
For $\dim X = k$, we define the **plurigenus**
\[
P_m(X) \da H^0( (\Omega^k_X)^{\tensor m})
.\]
For $m=1$, this is referred to as the **geometric genus**.
:::

:::{.theorem}
(Castelnuovo's rationality criterion) If $X$ is a smooth projective surface, then $X$ is rational iff

- $h^{1, 0}(X) = 0$
- $P_2(X) = 0$

:::