Geodesic
Effective results in the theory of birational rigidity
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 2, pp. 301-354 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper is a survey of recent effective results in the theory of birational rigidity of higher-dimensional Fano varieties and Fano–Mori fibre spaces. Bibliography: 59 titles.
Keywords: Fano variety, birational rigidity, linear system, maximal singularity, quadratic singularity, multiquadratic singularity.
Mots-clés : Mori fibre space, birational map 
@article{RM_2022_77_2_a2,
     author = {A. V. Pukhlikov},
     title = {Effective results in the theory of birational rigidity},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {301--354},
     year = {2022},
     volume = {77},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2022_77_2_a2/}
}
TY  - JOUR
AU  - A. V. Pukhlikov
TI  - Effective results in the theory of birational rigidity
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2022
SP  - 301
EP  - 354
VL  - 77
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/RM_2022_77_2_a2/
LA  - en
ID  - RM_2022_77_2_a2
ER  - 
%0 Journal Article
%A A. V. Pukhlikov
%T Effective results in the theory of birational rigidity
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2022
%P 301-354
%V 77
%N 2
%U http://geodesic.mathdoc.fr/item/RM_2022_77_2_a2/
%G en
%F RM_2022_77_2_a2
A. V. Pukhlikov. Effective results in the theory of birational rigidity. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 2, pp. 301-354. http://geodesic.mathdoc.fr/item/RM_2022_77_2_a2/

[1] C. Birkar, P. Cascini, C. D. Hacon, J. McKernan, “Existence of minimal models for varieties of log general type”, J. Amer. Math. Soc., 23:2 (2010), 405–468 | DOI | MR | Zbl

[2] V. A. Iskovskikh, “Birational rigidity of Fano hypersurfaces in the framework of Mori theory”, Russian Math. Surveys, 56:2 (2001), 207–291 | DOI | DOI | MR | Zbl

[3] I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965 | DOI | DOI | MR | Zbl

[4] A. V. Pukhlikov, “Birationally rigid varieties. I. Fano varieties”, Russian Math. Surveys, 62:5 (2007), 857–942 | DOI | DOI | MR | Zbl

[5] A. V. Pukhlikov, “Birationally rigid varieties. II. Fano fibre spaces”, Russian Math. Surveys, 65:6 (2010), 1083–1171 | DOI | DOI | MR | Zbl

[6] A. Pukhlikov, Birationally rigid varieties, Math. Surveys Monogr., 190, Amer. Math. Soc., Providence, RI, 2013, vi+365 pp. | DOI | MR | Zbl

[7] A. V. Pukhlikov, “Birational geometry of higher-dimensional Fano varieties”, Proc. Steklov Inst. Math., 288, suppl. 2 (2015), S1–S150 | DOI | DOI | MR | Zbl

[8] T. de Fernex, “Birationally rigid hypersurfaces”, Invent. Math., 192:3 (2013), 533–566 ; “Erratum”, 203:2 (2016), 675–680 | DOI | MR | Zbl | DOI | MR

[9] F. Suzuki, “Birational rigidity of complete intersections”, Math. Z., 285:1-2 (2017), 479–492 | DOI | MR | Zbl

[10] Ziquan Zhuang, “Birational superrigidity and $K$-stability of Fano complete intersections of index 1”, Duke Math. J., 169:12 (2020), 2205–2229 | DOI | MR | Zbl

[11] Th. Eckl, A. Pukhlikov, “On the locus of nonrigid hypersurfaces”, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat., 79, Springer, Cham, 2014, 121–139 | DOI | MR | Zbl

[12] A. V. Pukhlikov, “Birationally rigid Fano fibre spaces. II”, Izv. Math., 79:4 (2015), 809–837 | DOI | DOI | MR | Zbl

[13] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA2), Augmenté d'un exposé par M. Raynaud. Séminaire de géométrie algébrique du Bois-Marie, 1962, Adv. Stud. Pure Math., 2, North-Holland Publishing Co., Amsterdam; Masson Cie, Editeur, Paris, 1968, vii+287 pp. | MR | Zbl

[14] F. Call, G. Lyubeznik, “A simple proof of Grothendieck's theorem on the parafactoriality of local rings”, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994, 15–18 | DOI | MR | Zbl

[15] F. W. Call, “A theorem of Grothendieck using Picard groups for the algebraist”, Math. Scand., 74:2 (1994), 161–183 | DOI | MR | Zbl

[16] A. V. Pukhlikov, “Birationally rigid Fano complete intersections”, J. Reine Angew. Math., 2001:541 (2001), 55–79 | DOI | MR | Zbl

[17] A. V. Pukhlikov, “Birational automorphisms of Fano hypersurfaces”, Invent. Math., 134:2 (1998), 401–426 | DOI | MR | Zbl

[18] A. V. Pukhlikov, “The $4n^2$-inequality for complete intersection singularities”, Arnold Math. J., 3:2 (2017), 187–196 | DOI | MR | Zbl

[19] V. V. Shokurov, “3-fold log flips”, Russian Acad. Sci. Izv. Math., 40:1 (1993), 95–202 | DOI | MR | Zbl

[20] J. Kollár (ed.), Flips and abundance for algebraic threefolds (Univ. of Utah, Salt Lake City, 1991), Astérisque, 211, Soc. Math. France, Paris, 1992, 258 pp. | MR | Zbl

[21] A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. Math., 66:6 (2002), 1243–1269 | DOI | DOI | MR | Zbl

[22] A. V. Pukhlikov, “Birational geometry of algebraic varieties with a pencil of Fano complete intersections”, Manuscripta Math., 121:4 (2006), 491–526 | DOI | MR | Zbl

[23] A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces with isolated singularities”, Sb. Math., 193:3 (2002), 445–471 | DOI | DOI | MR | Zbl

[24] A. V. Pukhlikov, “Birationally rigid complete intersections with a singular point of high multiplicity”, Proc. Edinb. Math. Soc. (2), 62:1 (2019), 221–239 | DOI | MR | Zbl

[25] D. Foord, “Birationally rigid Fano cyclic covers over a hypersurface containing a singular point”, Eur. J. Math., 2020, 1–16, Publ. online | DOI

[26] Th. Eckl, A. Pukhlikov, “Effective birational rigidity of Fano double hypersurfaces”, Arnold Math. J., 4:3-4 (2018), 505–521 | DOI | MR | Zbl

[27] D. Evans, A. V. Pukhlikov, “Birationally rigid complete intersections of high codimension”, Izv. Math., 83:4 (2019), 743–769 | DOI | DOI | MR | Zbl

[28] A. V. Pukhlikov, “Birational geometry of varieties, fibred into complete intersections of codimension two”, Izv. Math., 86:2 (2022) ; (2021), 86 pp., arXiv: 2101.10830 | DOI

[29] A. V. Pukhlikov, “Birationally rigid finite covers of the projective space”, Proc. Steklov Inst. Math., 307 (2019), 232–244 | DOI | DOI | MR | Zbl

[30] A. V. Pukhlikov, “Canonical and log canonical thresholds of multiple projective spaces”, Eur. J. Math., 7:1 (2021), 135–162 ; (2019), 27 pp., arXiv: 1906.11802 | DOI | MR | Zbl

[31] V. A. Iskovskih, Yu. I. Manin, “Three-dimensional quartics and counterexamples to the Lüroth problem”, Math. USSR-Sb., 15:1 (1971), 141–166 | DOI | MR | Zbl

[32] A. V. Pukhlikov, “Birational geometry of algebraic varieties fibred into Fano double spaces”, Izv. Math., 81:3 (2017), 618–644 | DOI | DOI | MR | Zbl

[33] A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. Math., 69:6 (2005), 1225–1255 | DOI | DOI | MR | Zbl

[34] A. V. Pukhlikov, “Rationally connected rational double covers of primitive Fano varieties”, Épijournal Géom. Algébrique, 4 (2020), 18, 14 pp. | DOI | MR | Zbl

[35] A. V. Pukhlikov, “Remarks on Galois rational coverings”, Math. Notes, 110:2 (2021), 242–247 | DOI | DOI | MR | Zbl

[36] J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, Berlin, 1996, viii+320 pp. | DOI | MR | Zbl

[37] A. V. Pukhlikov, “Birational geometry of Fano hypersurfaces of index two”, Math. Ann., 366:1-2 (2016), 721–782 | DOI | MR | Zbl

[38] A. V. Pukhlikov, “Birational geometry of singular Fano hypersurfaces of index two”, Manuscripta Math., 161:1-2 (2020), 161–203 | DOI | MR | Zbl

[39] A. V. Pukhlikov, “Birational geometry of Fano double spaces of index two”, Izv. Math., 74:5 (2010), 925–991 | DOI | DOI | MR | Zbl

[40] A. V. Pukhlikov, “Birational geometry of singular Fano double spaces of index two”, Sb. Math., 212:4 (2021), 551–566 | DOI | DOI | MR | Zbl

[41] J.-L. Colliot-Thélène, A. Pirutka, “Cyclic covers that are not stably rational”, Izv. Math., 80:4 (2016), 665–677 | DOI | DOI | MR | Zbl

[42] B. Hassett, A. Kresch, Y. Tschinkel, “Stable rationality and conic bundles”, Math. Ann., 365:3-4 (2016), 1201–1217 | DOI | MR | Zbl

[43] B. Totaro, “Hypersurfaces that are not stably rational”, J. Amer. Math. Soc., 29:3 (2016), 883–891 | DOI | MR | Zbl

[44] A. Auel, Ch. Böhning, A. Pirutka, “Stable rationality of quadric and cubic surface bundle fourfolds”, Eur. J. Math., 4:3 (2018), 732–760 | DOI | MR | Zbl

[45] B. Hassett, A. Pirutka, Yu. Tschinkel, “A very general quartic double fourfold is not stably rational”, Algebr. Geom., 6:1 (2019), 64–75 | DOI | MR | Zbl

[46] S. Schreieder, “Stably irrational hypersurfaces of small slopes”, J. Amer. Math. Soc., 32:4 (2019), 1171–1199 | DOI | MR | Zbl

[47] Yu. Prokhorov, C. Shramov, “Jordan property for groups of birational selfmaps”, Compos. Math., 150:12 (2014), 2054–2072 | DOI | MR | Zbl

[48] Yu. Prokhorov, C. Shramov, “Jordan property for Cremona groups”, Amer. J. Math., 138:2 (2016), 403–418 | DOI | MR | Zbl

[49] A. Avilov, “Automorphisms of singular three-dimensional cubic hypersurfaces”, Eur. J. Math., 4:3 (2018), 761–777 | DOI | MR | Zbl

[50] A. A. Avilov, “Biregular and birational geometry of quartic double solids with 15 nodes”, Izv. Math., 83:3 (2019), 415–423 | DOI | DOI | MR | Zbl

[51] I. Cheltsov, A. Dubouloz, J. Park, “Super-rigid affine Fano varieties”, Compos. Math., 154:11 (2018), 2462–2484 | DOI | MR | Zbl

[52] A. Dubouloz, T. Kishimoto, “Explicit biregular/birational geometry of affine threefolds: completions of ${\mathbb A}^3$ into del Pezzo fibrations and Mori conic bundles”, Algebraic varieties and automorphism groups, Adv. Stud. Pure Math., 75, Math. Soc. Japan, Tokyo, 2017, 49–71 | DOI | MR | Zbl

[53] I. Cheltsov, V. Przyjalkowski, C. Shramov, “Quartic double solids with icosahedral symmetry”, Eur. J. Math., 2:1 (2016), 96–119 | DOI | MR | Zbl

[54] I. Cheltsov, V. Przyjalkowski, C. Shramov, “Burkhardt quartic, Barth sextic, and the icosahedron”, Int. Math. Res. Not. IMRN, 2019:12 (2019), 3683–3703 | DOI | MR | Zbl

[55] I. Cheltsov, C. Shramov, “Three embeddings of the Klein simple group into the Cremona group of rank three”, Transform. Groups, 17:2 (2012), 303–350 | DOI | MR | Zbl

[56] I. Cheltsov, C. Shramov, “Five embeddings of one simple group”, Trans. Amer. Math. Soc., 366:3 (2014), 1289–1331 | DOI | MR | Zbl

[57] I. Cheltsov, C. Shramov, Cremona groups and the icosahedron, Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2016, xxi+504 pp. | DOI | MR | Zbl

[58] Yu. Prokhorov, C. Shramov, “Finite groups of birational selfmaps of threefolds”, Math. Res. Lett., 25:3 (2018), 957–972 | DOI | MR | Zbl

[59] Yu. Prokhorov, C. Shramov, “$p$-subgroups in the space Cremona group”, Math. Nachr., 291:8–9 (2018), 1374–1389 | DOI | MR | Zbl