Jacobian variety

In mathematics, a Jacobian variety is a special kind of space that helps us understand certain types of equations and shapes. It is connected to what we call an algebraic curve, which is a smooth and continuous shape that can be described using equations.

The Jacobian variety is built from something called line bundles, which are ways to add and arrange certain features on the curve. These line bundles have a degree of zero, meaning they balance out in a particular way.

Jacobian varieties are important because they give us a way to study the properties of these curves. They are also linked to other areas of math, like abelian varieties, which are special types of geometric objects with nice symmetry. This makes Jacobian varieties a key idea in many parts of modern mathematics.

Introduction

The Jacobian variety is named after Carl Gustav Jacobi, who helped prove an important math idea called the Abel–Jacobi theorem. This idea connects the work of Niels Abel. The Jacobian variety is a special kind of math object called an abelian variety, and it has a size called dimension g. When we work with complex numbers, it looks like a complex torus. If we pick a point p on a curve C, we can connect C to a smaller part of the Jacobian variety, with p becoming the starting point, and C helps create the whole Jacobian variety as a group.

Construction for complex curves

Over the complex numbers, the Jacobian variety can be seen as a special kind of space made by dividing one space by another. This helps us understand how certain mathematical objects are put together.

This idea was expanded by mathematicians working with curves over other types of fields, helping solve important problems in number theory.

Algebraic structure

The Jacobian variety of a curve is a special group. It is connected to a set of special math objects called divisors that have a degree of zero. These divisors come from rational functions, and this idea works well even when we look at fields that are not algebraically closed.

Further notions

Torelli's theorem says that a complex curve can be identified by its Jacobian, which includes information about how it is polarized.

The Schottky problem asks which special types of abelian varieties come from curves. There are also ideas like the Picard variety, Albanese variety, generalized Jacobian, and intermediate Jacobians, which extend the idea of the Jacobian to higher-dimensional shapes. For higher dimensions, the way we build the Jacobian using holomorphic 1-forms leads to the Albanese variety, though it may not always match the Picard variety.