Birational geometry

Birational geometry is a branch of algebraic geometry that studies the relationships between algebraic varieties through birational equivalences. These are equivalences that allow the objects in question to be related by rational maps, which can typically be viewed as fewer-dimensional representations of the varieties.

In algebraic geometry, a **birational invariant** is a property of a variety (or more generally, an algebraic scheme) that remains unchanged under birational equivalence. Two varieties \( X \) and \( Y \) are said to be birationally equivalent if there exist rational maps from \( X \) to \( Y \) and from \( Y \) to \( X \) that are inverses of each other on a dense open subset of each variety.

"Blowing up" can refer to a variety of contexts and meanings depending on the subject matter. Here are a few common interpretations: 1. **Explosions**: In a literal sense, "blowing up" can refer to something exploding or bursting apart, such as a bomb or a balloon. 2. **Popularity/Success**: In a figurative sense, especially in social media or entertainment, "blowing up" means achieving sudden and significant success or widespread recognition.

An elliptic surface is a type of algebraic surface that has a fibration structure, meaning it can be viewed as a family of elliptic curves. In more technical terms, an elliptic surface is a smooth projective surface \(S\) over a base scheme, typically taken to be the complex numbers, which admits a morphism to a base scheme \(B\) such that for every point in \(B\), the fiber over that point is an elliptic curve.

An Enriques surface is a specific type of algebraic surface that has several interesting geometric and topological properties. They are named after the Italian mathematician Federigo Enriques, who studied these types of surfaces in the early 20th century. Here are some key characteristics and properties of Enriques surfaces: 1. **Classification**: Enriques surfaces belong to a broader classification of surfaces in algebraic geometry, which includes other types like K3 surfaces, rational surfaces, and so on.

The Enriques–Kodaira classification is a fundamental classification scheme in the field of algebraic geometry that categorizes compact complex surfaces based on their geometric properties. It was developed by the mathematicians Francesco Enriques and Katsumi Kodaira. The classification divides compact complex surfaces into several types, primarily based on their topological and geometric characteristics, particularly their canonical bundles.

The Iitaka dimension is a concept from algebraic geometry, specifically in the study of algebraic varieties and their properties. It is named after Shigeharu Iitaka, who introduced the notion. The Iitaka dimension of a projective variety (or more generally, a proper algebraic variety) is a measure of the growth rate of global sections of line bundles on the variety.

A rational surface is a type of algebraic surface that can be defined over an algebraically closed field and can be parametrized by rational functions.