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The rationality problem for conic bundles

Yu. G. Prokhorov<sup>ab

a</sup> Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
^b Laboratory of Algebraic Geometry and its Applications, National Research University Higher School of Economics

Abstract: This expository paper is concerned with the rationality problem for three-dimensional algebraic varieties with a conic bundle structure. The main methods of this theory are discussed, proofs of certain principal results are sketched, and some recent achievements are presented. Many open problems are also stated.
Bibliography: 209 titles.

Keywords: algebraic threefolds, conic bundles, rationality, singularities, invariants, birational transformations, Sarkisov programme, intermediate Jacobian, Prym variety.

UDC: 512.776

MSC: Primary 14E05, 14J30; Secondary 14E07, 14E30, 14J45

Received: 15.12.2017
Revised: 22.03.2018

DOI: 10.4213/rm9811

 English version: Russian Mathematical Surveys, 2018, 73:3, 375–456

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ArXiv: 1712.05564