---
date: 2022-01-15 21:49
modification date: Friday 28th January 2022 23:06:27
title: curves
aliases: [curve]

---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #AG #arithmetic-geometry 
- Refs:
	- #todo/add-references
- Links:
	- [scheme](Unsorted/scheme.md)
	- [elliptic curve](MOCs/elliptic%20curve.md)
	- [algebraic curve](Unsorted/algebraic%20curve.md)
	- [good reduction](good%20reduction)
	- [semistable reduction](semistable%20reduction)

---

# curves

# Definitions

- **Variety**: an [integral](Unsorted/integrally%20closed.md)  [separated](Unsorted/separated.md) [scheme](scheme.md) of [finite type](finite%20type.md) over $k$.
- **Dimension**: dimension as a Noetherian topological space.
- **Curve**: a [complete variety](complete%20variety) (proper over $k$) of dimension 1.
- **Singular**: for $\mfm \normal \OO_{X, x}$ the maximal ideal of the local ring at a closed point $x$, $X$ is singular if $\dim_{\kappa(x)} \mfm/\mfm^2 \neq 1$ where $\kappa(x)$ is the residue field at $x$. 
- [valuation](Unsorted/Valuations.md) : for $f \in \OO_{X, x}$, $v_p(f)$ is the largest $n$ such that $f\in \mfm^n$.
	- Zero: $v_p(f)  > 0$
	- Pole: $v_p(f) < 0$
	- Nonvanishing: $v_p(f) = 0$.
- [Unsorted/function field](Unsorted/function%20field.md) : the local ring $K(X) \da \OO_{X, \tilde x}$ for $\tilde x$ the generic point.
- **Degree**: for $f:X\to Y$, the degree of the field extension $[K(X) : f^* K(Y)]$.
- [Ramification index](Ramification%20index.md) : $e_f(x)$ defined as the largest $n$ such that $f^* \mfm_{f(x)} \subseteq \mfm_x^n$.
- [ramified](Unsorted/ramification%20index.md) : $e_f(x) > 1$.
	- Alternatively: at a closed point $x$, $f^* \mfm_{f(x)} = \mfm_x$ and the extension $\OO_x/\mfm_x$ is a finite separable extension of $\OO_{f(x)} / \mfm_{f(x)}$.
- Structure morphism: for a scheme $X$ over $k$, the map $S: X\to \spec k$
- [geometric point](Unsorted/geometric%20fiber.md) : a section to the structure map, $s: \spec k \to X$ so that $\spec k \mapsvia{s} X \mapsvia{S} \spec k$ is the identity on $\spec k$
- [elliptic curve](elliptic%20curve.md) : genus 1. Coincides with $y^2 = x^3 + Ax  + B$. Exists as a pointed [scheme](scheme.md) $(E, O)$
- [isogeny](isogeny) : a pointed map $(E_1, O_1) \mapsvia{f} (E_2, O_2)$, so $f(O_1) = O_2$. 
- [supersingular](MOCs/elliptic%20curve.md) : $\ker\qty{E \mapsvia{\cdot p} E } = 0$ where $\ch k = p > 0$.
- **Ordinary**: $\ker\qty{E\mapsvia{\cdot p} E } = \ZZ/p$
- [etale morphism](Unsorted/etale.md) : [flat](Unsorted/faithfully%20flat.md) and [unramified](Unsorted/unramified.md). Supposed to look like a local homeomorphism in a covering space.

![](attachments/Pasted%20image%2020220407190651.png)

# Genus formula

![](attachments/Pasted%20image%2020220214101151.png)

# Examples

## Genus 0

#todo 

## Genus 1
[elliptic curves](MOCs/elliptic%20curve.md)

## Genus 2
[hyperellptic](Unsorted/hyperellptic.md)

![](attachments/Pasted%20image%2020220214104026.png)

## Genus 3

![](attachments/Pasted%20image%2020220214104810.png)
![](attachments/Pasted%20image%2020220214104904.png)

## Genus 4
![](attachments/Pasted%20image%2020220214104930.png)

## Genus 5

![](attachments/Pasted%20image%2020220214104946.png)

# Homology

![](attachments/Pasted%20image%2020220214113837.png)

# Examples

![](attachments/Pasted%20image%2020220410185146.png)