Abstract:
The theory of birational rigidity of rationally connected varieties generalises the classical rationality problem. This paper gives a survey of the current state of this theory and traces its history from Noether's theorem and the Lüroth problem to the latest results on the birational superrigidity of higher-dimensional Fano varieties. The main components of the method of maximal singularities are considered.
This publication is cited in the following 17 articles:
A. V. Pukhlikov, “Effective results in the theory of birational rigidity”, Russian Math. Surveys, 77:2 (2022), 301–354
Pukhlikov A.V., “Birational geometry of Fano hypersurfaces of index two”, Math. Ann., 366:1-2 (2016), 721–782
Beauville A., “The Luroth Problem”, Rationality Problems in Algebraic Geometry, Lect. Notes Math., Lecture Notes in Mathematics, 2172, ed. Pardini R. Pirola G., Springer International Publishing Ag, 2016, 1–27
T. Eckl, A. Pukhlikov, “On the locus of nonrigid hypersurfaces”, Springer Proc. Math. Statist., 79 (2014), 121–139
A. V. Pukhlikov, “Birational rigidity of Fano complete intersections”, Dokl. Math, 87:1 (2013), 34
A. V. Pukhlikov, “Birationally rigid complete intersections of quadrics and cubics”, Izv. Math., 77:4 (2013), 795–845
Yuji Odaka, Takuzo Okada, “Birational superrigidity and slope stability of Fano manifolds”, Math. Z, 2013
Kaloghiros A.-S., “A classification of terminal quartic 3-folds and applications to rationality questions”, Math. Ann., 354:1 (2012), 263–296
R. J. Mullany, “Fano Double Spaces with a Big Singular Locus”, Math. Notes, 87:3 (2010), 444–448
A. V. Pukhlikov, “Birational geometry of Fano double spaces of index two”, Izv. Math., 74:5 (2010), 925–991
A. V. Pukhlikov, “Birationally rigid varieties. II. Fano fibre spaces”, Russian Math. Surveys, 65:6 (2010), 1083–1171
A. V. Pukhlikov, “Birational Geometry of Singular Fano Varieties”, Proc. Steklov Inst. Math., 264 (2009), 159–177
F. A. Bogomolov, Vik. S. Kulikov, Yu. I. Manin, V. V. Nikulin, D. O. Orlov, A. N. Parshin, Yu. G. Prokhorov, A. V. Pukhlikov, M. Reid, I. R. Shafarevich, V. V. Shokurov, “Vasilii Alekseevich Iskovskikh (obituary)”, Russian Math. Surveys, 64:5 (2009), 939–946
A. V. Pukhlikov, “Singularities of Algebraic Subvarieties and Problems of Birational Geometry”, Proc. Steklov Inst. Math., 267 (2009), 235–247
A. V. Pukhlikov, “Birational geometry of Fano double covers”, Sb. Math., 199:8 (2008), 1225–1250
A. V. Pukhlikov, “On the self-intersection of a movable linear system”, J. Math. Sci., 164:1 (2010), 119–130
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