The rationality problem for conic bundles

TL;DR

This paper explores the rationality problem for three-dimensional algebraic varieties with conic bundle structures, discussing key methods, recent results, and open questions in the field.

Contribution

It provides an overview of main methods, sketches proofs of key results, and highlights recent advances and open problems in the rationality of conic bundles.

Findings

Discussion of main methods in the theory

Sketches of proofs for principal results

Presentation of recent achievements and open problems

Abstract

This expository paper is concerned with the rationality problems for three-dimensional algebraic varieties with a conic bundle structure. We discuss the main methods of this theory. We sketch the proofs of certain principal results, and present some recent achievements. Many open problems are also stated.