# Lecture 1: Discussion and Review

## Valuations

- Transcendence bases
- Lüroth problem

For $K/k$ a one variable function field, if we want a curve $C/k$, what are the points?
We'll use *valuations*, see NT 2.1.
See also completions, residue fields.
If $R \subset K$ a field, $R$ is a *valuation ring* of $K$ if for all $x\in K\units$, at least one of $x, x^{-1} \in R$.

:::{.example}
The valuation rings of $\QQ$ are $\ZZ_{(p)}\da \ZZ[\ts{{1\over \ell} \st \ell\neq p}]$ for all primes $p$.
:::

See also *Krull valuations*, which take values in some totally ordered commutative group.

:::{.exercise}
Show that a valuation ring is a local ring, i.e. it has a unique maximal ideal.
:::

:::{.example}
Where does the log come from?

There is a $p\dash$adic valuation:
\[  
v: \QQ &\to \ZZ_{(p)} \\
{a\over b} = p^n {u \over v} &\mapsto n
.\]
 
Then we recover
\[  
\ZZ_{(p)} = \ts {x\in \QQ\units \st v_p(x) \geq 0} \union \ts{0} \\
\mfm_{(p)} = \ts {x\in \QQ\units \st v_p(x) > 0} \union \ts{0} \\
.\]

There is a $p\dash$adic norm
\[  
\abs{\wait}_p: \QQ &\to \RR \\
0 & \mapsto 0 \\
x &\mapsto p^{-n} = p^{-v_p(x)}
.\]

Then we get an ultrametric function, a non-archimedean function
\[  
d_p: \QQ^2 \to \RR \\
(x, y) &\mapsto \abs{x- y}_p
.\]

We then recover $v_p(x) = -\log_p \abs{x}_p$.[^See_NT1]

[^See_NT1]: 
See NT 1 notes for more details on valuations.

:::


## Places

For $A\subset K$ a subring of a field, we'll be interested in the place $\tilde \Sigma = \ts{\text{Valuation rings } R_v \text{ of } K} \st A \subset R_v \subsetneq K$.
Thus the valuation takes non-negative values on all elements of $K$.
Can equip this with a topology (the "Zariski" topology, not the usual one).
This is always quasicompact, and called the *Zariski-Riemann space*.
Can determine a sheaf of rings to make this a locally ringed space.

We can define an equivalence of valuations and define the set of *places* 
\[  
\Sigma(K/k) \da \ts{\text{Nontrivial valuations } v\in K \st v(x) \geq 0\, \forall x\in k\units}
,\]
which will be the points on the curve.
Here the Zariski topology will be the cofinite topology (which is not Hausdorff).
Scheme-theoretically, this is exactly the set of closed points on the curve.

:::{.definition title="Generic Points"}
A point $p\in X$ a topological space is a **generic point** iff its closure in $X$ is all of $X$.
:::

:::{.remark}
Note we will have unique models for curves, but this won't be the case for surfaces: blowing up a point will yield a birational but inequivalent surface.
:::

## Divisors

:::{.definition title="Divisor Group"}
The *divisor group* of $K$ is the free $\ZZ\dash$module on $\Sigma(K/k)$
:::

:::{.remark}
This comes with a degree map
\[  
\deg: \Div(K) \to \ZZ
\]
which need not be surjective.
:::

:::{.definition title="Principal Divisors"}
Consider the map
\[  
\phi_d: K\units &\to \Div(K) \\
f &\mapsto (f)
.\]
Then we define $\im \phi_d$ as the subgroup of **principal divisors**.
:::


:::{.definition title="Class Group"}
Define the **class group** of $K$ as 
\[  
\cl(K) \da \ts{\text{Divisors}} / \ts{\text{Principal divisors}} \da \Div(K) / \im \phi_d
.\]
:::

We can define the **class group** as divisors modulo principle divisors $\cl(K) = \Div(K) / \im(K\units)$ and the Riemann-Roch space $\mathcal{L}(D)$.
The Riemann-Roch theorem will then be a statement about $\dim \mathcal{L}(D)$.