Yu. G. Prokhorov, “The rationality problem for conic bundles”, Russian Math. Surveys, 73:3 (2018), 375

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Russian Mathematical Surveys, 2018, Volume 73, Issue 3, Pages 375–456
DOI: https://doi.org/10.1070/RM9811 (Mi rm9811)

This article is cited in 25 scientific papers (total in 25 papers)

The rationality problem for conic bundles

Yu. G. Prokhorov^ab

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

English version PDF (1356 kB) Citations (25) Russian version article

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DOI: https://doi.org/10.1070/RM9811

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.

Funding agency	Grant number
Ministry of Education and Science of the Russian Federation	5-100
This work was partially supported by the Russian Academic Excellence Project ``5-100''. The paper was written while the author was visiting the Max Planck Institute for Mathematics (Bonn). He would like to thank the institute for the invitation and the excellent working conditions.

Received: 15.12.2017
Revised: 22.03.2018

Bibliographic databases:

Document Type: Article

UDC: 512.776

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

Language: English

Original paper language: Russian

Citation: Yu. G. Prokhorov, “The rationality problem for conic bundles”, Russian Math. Surveys, 73:3 (2018), 375–456

Citation in format AMSBIB

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Linking options:

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https://doi.org/10.1070/RM9811

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Related presentations:

Проблема рациональности для особых многообразий Фано
Yu. G. Prokhorov, November 23, 2022 12:20

This publication is cited in the following 25 articles:

Nicholas Lindsay, “On 3-folds having a holomorphic torus action with 6 fixed points”, Journal of Geometry and Physics, 2025, 105433
Alessio Corti, Giulia Gugiatti, Fernando Rodriguez Villegas, Progress in Mathematics, 351, Perspectives on Four Decades of Algebraic Geometry, Volume 1, 2025, 185
Lena Ji, Mattie Ji, “Rationality of Real Conic Bundles With Quartic Discriminant Curve”, International Mathematics Research Notices, 2024:1 (2024), 115
Ivan Cheltsov, Elena Denisova, Kento Fujita, “K-stable smooth Fano threefolds of Picard rank two”, Forum of Mathematics, Sigma, 12 (2024)
Ciro Ciliberto, “On the rationality of certain Fano threefolds”, manuscripta math., 174:1-2 (2024), 203
Shin-ichi Matsumura, Xiaojun Wu, “Compact Kähler three-folds with nef anti-canonical bundle”, Math. Ann., 2024
Tatiana Bandman, Yuri G. Zarhin, “Jordan Groups and Geometric Properties of Manifolds”, Arnold Math J., 2024
Renato Dias Costa, “Classification of conic bundles on a rational elliptic surface in any characteristic”, Beitr Algebra Geom, 2024
HOSUNG KIM, JEONG-SEOP KIM, YONGNAM LEE, “BIGNESS OF THE TANGENT BUNDLE OF A FANO THREEFOLD WITH PICARD NUMBER TWO”, Nagoya Math. J., 2024, 1
I. Cheltsov, K. Fujita, T. Kishimoto, T. Okada, “K-stable divisors in $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ of degree $(1,1,2)$ ”, Nagoya Math. J., 251 (2023), 686–714
J. Blanc, I. Cheltsov, A. Duncan, Yu. Prokhorov, “Finite quasisimple groups acting on rationally connected threefolds”, Math. Proc. Camb. Phil. Soc., 174:3 (2023), 531
Yu. Prokhorov, “Rationality of Fano threefolds with terminal Gorenstein singularities, II”, Rend. Circ. Mat. Palermo, II. Ser., 72:3 (2023), 1797
Alex Massarenti, “On the unirationality of quadric bundles”, Advances in Mathematics, 431 (2023), 109235
Yu. G. Prokhorov, “Trekhmernye mnogoobraziya Fano”, Lekts. kursy NOTs, 31, MIAN, M., 2022, 3–154
C. Ciliberto, M. Zaidenberg, “Lines, conics, and all that”, Pure Appl. Math. Q., 18:1 (2022), 101–176
Yu. Prokhorov, “Conic bundle structures on $\mathbb{Q}$ -Fano threefolds”, Electronic Research Archive, 30:5 (2022), 1881–1897
Jingjun Han, Chen Jiang, Yujie Luo, “Shokurov's conjecture on conic bundles with canonical singularities”, Forum of Mathematics, Sigma, 10 (2022)
Yu. G. Prokhorov, “Equivariant minimal model program”, Russian Math. Surveys, 76:3 (2021), 461–542
I. Cheltsov, J. Park, Yu. Prokhorov, M. Zaidenberg, “Cylinders in Fano varieties”, EMS Surv. Math. Sci., 8:1-2 (2021), 39–105
Pedro Montero, Eleonora Anna Romano, “A Characterization of Some Fano 4-folds Through Conic Fibrations”, 2021, no. 16, 2021, 12009

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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Abstract page:	1068
Russian version PDF:	188
English version PDF:	134
References:	100
First page:	46

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