# Thursday, February 02

:::{.theorem title="Riemann-Hurwitz"}
If $f:C\to D$ is a map of smooth complete curves then
\[
2g(C) - 2 = \deg(f)\cdot\qty{ 2g(D) - 2 } + \deg R(f)
\]
where $R(f) = \sum_{p \in C} (e_p - 1)[p]$ is the ramification divisor.
:::

:::{.example title="?"}
Take 
\[
\CC &\to \CC \\
z &\mapsto z^{4}
\]
which is a 4-fold cover

\begin{tikzpicture}
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\end{tikzpicture}

:::

:::{.example title="?"}
If $f: C\to \PP^1$ is degree 2 then $f=a/b$ where $a,b\in H^0(C; K_C)$, and Riemann-Hurwitz gives $\deg R(f) = 6$.
If $\tv{s: t}$ are homogeneous coordinates on $\PP^1$, then one can take an equation of the form $z^2 = f_6(s, t)$ for $f_6(s,t) = \prod_{i=1}^6 (a_i s - b_i t)$ is a homogeneous degree 6 polynomial, so $f\in H^0(\PP^1;\OO_{\PP^1}(6))$ and $z\in \Tot \OO_{\PP^1}(-3)$: 

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-02-02_09-59.pdf_tex} };
\end{tikzpicture}

:::

:::{.exercise title="?"}
Describe why $z\not\in \Tot \OO_{\PP^1}(3)$ instead.
:::

:::{.exercise title="?"}
Recall that the normalization of a ring $R$ is the integral closure of $R$ in $\ff(R)$.
Compute the normalization of $y^2=x^3$ using the algebraic definition.
:::

:::{.example title="?"}
An example of normalization:

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\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Spring/k3surfaces/sections/figures}{2023-02-02_10-22.pdf_tex} };
\end{tikzpicture}

:::

:::{.fact}
Integrally closed and 1-dimensional implies smooth.
:::

:::{.exercise title="?"}
Recall the definition of the Čech cochain complex and compute $\Hc(S^1; \mcf)$ using an open cover of two sets.
:::