How Do You Spell BIRATIONAL GEOMETRY?

1. Birational geometry is a branch of mathematics that studies the geometry of algebraic varieties through birational transformations. It deals with the classification and comparison of algebraic varieties, particularly focusing on varieties that are closely related through rational maps.

   In algebraic geometry, an algebraic variety is a geometric object defined by the solution set of algebraic equations over a field, usually the complex numbers. A birational transformation is a rational map that has an inverse, which means it preserves the birational equivalence between varieties.

   The central goal of birational geometry is to understand and classify the birational properties of algebraic varieties. Two algebraic varieties are said to be birational if there exists a rational map between them that can be inverted. Birational transformations capture the intrinsic similarity between varieties and provide a way to study them by comparing birationally equivalent varieties.

   By studying birational transformations, birational geometry aims to answer important questions about algebraic varieties such as their singularities, geometric properties, and moduli spaces. It investigates how the geometry and properties of a variety change under birational transformations, shedding light on the interplay between the different aspects of the variety.

   Birational geometry has applications in various branches of mathematics, including number theory, complex analysis, and algebraic topology. It plays a crucial role in understanding the geometry of higher-dimensional algebraic varieties and provides tools and techniques for resolving fundamental questions in algebraic geometry.

The word "birational" in "birational geometry" comes from the mathematics concept of birational maps. The term "biration" is a combination of "bi-" meaning two and "ration" meaning ratio, implying a ratio of two things. In mathematics, a birational map is a rational map (a map defined by rational functions) that has an inverse rational map, meaning that there is a correspondence between two algebraic varieties that is given by rational functions.

The term "birational geometry" itself refers to the study of algebraic varieties (geometric objects defined by algebraic equations) from a birational perspective. It involves understanding the geometric properties and transformations of these varieties based on the concept of birational maps. The term "birational geometry" was coined to describe this specific approach and field of research within algebraic geometry.