---
date: 2022-04-06 04:42
modification date: Saturday 6th August 2022 17:45:46
title: "Jacobian"
aliases: [Jacobian]
---

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

---

- Tags: 
	- #AG/moduli-spaces 
- Refs:
	- #todo/add-references
- Links:
	- [Abel-Jacobi theorem](Abel-Jacobi%20theorem)
	- [Torelli](Unsorted/Torelli.md)

---


# Jacobian

For $C$ a nonsingular [algebraic curve](algebraic%20curve.md), $\Jac(C)$ is the connected component of the identity in the [Picard group](Unsorted/Picard%20group.md) $\Pic(C)$, i.e. the moduli space of degree 0 line bundles on $C$.
Over $\CC$, can be realized as $\Jac(C) \cong H^0(X; \Omega^1_{C}) \dual/ H^1(X; \OO_X)$, where the embedding $H^1\embeds H^0$ uses [theta functions](theta%20function).

![](attachments/Pasted%20image%2020220806174542.png)