# Wednesday, September 13

## Prime Divisors

:::{.remark}
A prime divisor on an irreducible variety $X$ is an irreducible subvariety of codimension 1.
For example, note $X = V(xy) \subseteq \AA^2$ is reducible, as $X = V(x) \union V(y)$ which are both proper closed subvarieties.
An example of a divisor: $X = \AA^1$ and $D = p$ a point corresponding to the ideal $(x-p)$.
More generally, prime divisors are

- For curves: points; 
- For surfaces: irreducible curves;
- For threefolds: irreducible surfaces.

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:::{.remark}
Define $\Div(X)$ as the free abelian group generated by prime divisors on $X$.
A Weil divisor is an element of $\Div(X)$.
E.g. $D_1 \da \ts{0} \subseteq \PP^1$ is a prime divisor, as is $D_2 \da \ts{\infty }$, and $D \da a D_1 + b D_2$ is a Weil divisor for any $a,b\in \ZZ$.
We say $D_i = \sum a_i D_i$ is effective if $a_i \geq 0$ for all $i$, and we write $D \geq 0$.
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:::{.remark}
Let $f\in \CC(X)$ be a nonzero rational function on $X$ and $D \subseteq X$ a prime divisor.
We write $v_D(f)$ for the order of vanishing of $f$ along $D$.
We claim that any $f\in \CC(X)^*$ can be written locally on an open subset $U$ intersecting $D$ as $f = g^n h$ where $g$ is a regular function on $X$ and $h$ is a regular function such that $\ro{h}{D} \not\equiv 0$.
We set $v_D(f) \da n$ for this $n$.
Here we assume $X$ is normal, thus smooth in codimension 1, so singularities are in codimension at most 2.
We say $D$ is generically smooth if there exist open sets $U$ locally such that $D$ is defined by the vanishing of $g$, a regular function on $U$.
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:::{.example title="?"}
Take $D = V(x) \subseteq \AA^2$ and $f = {x\over y^2} \in \CC(\AA^2)$, then writing $f = x {1\over y^2}$ we have $g = x$ and $v_D(f) = 1$.
Similarly if $f = x^2$ then $v_D(f) = 2$, and $v_D\qty{y^3\over x^5} = -5$.
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:::{.remark}
We define $\div(f) \da \sum_D v_D(f) D$, a sum over all prime divisors. If $f = x^3/y^2$ then $\div(f) = 3D_1 -2 D_2$ where $D_1 = V(x)$ and $D_2 = V(y)$.
We say a Weil divisor $D$ is principal if $D = \div(f)$ for some $f\in \CC(X)^*$.
Say two divisors are linearly equivalent if their difference is principal.
Set $\Div_0(X) \subseteq \Div(X)$ to be the subset of principal divisors, and $\Cl(X) \da \Div(X) / \Div_0(X)$, which are Weil divisors up to linear equivalence.
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:::{.example title="?"}
Find an example of a Weil divisor which is not principal.
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:::{.remark}
Define a Cartier divisor to be a Weil divisor which is locally principal: there is an open cover of $X$ for which $D$ restricts to a principal divisor on each open set.
Define $\CDiv(X) \subseteq \Div(X)$ as the group of Cartier divisors, and $\Pic(X) \da \CDiv(X) / \CDiv_0(X)$, which are Cartier divisors up to linear equivalence.
If $X$ is smooth, $\Cl(X) \cong \Pic(X)$.
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