Algebraic hypersurfaces

Kollár, János

doi:10.1090/bull/1663

Algebraic hypersurfaces
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Bull. Amer. Math. Soc. 56 (2019), 543-568 Request permission

Abstract:

We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss these questions without any previous knowledge of algebraic geometry. At the end we formulate the main recent results and state the most important open questions.

References

S. S. Abhyankar, Lectures on expansion techniques in algebraic geometry, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay, 1977. Notes by Balwant Singh. MR 542446
E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
Carolina Araujo and János Kollár, Rational curves on varieties, Higher dimensional varieties and rational points (Budapest, 2001) Bolyai Soc. Math. Stud., vol. 12, Springer, Berlin, 2003, pp. 13–68. MR 2011743, DOI 10.1007/978-3-662-05123-8_{3}
Erik Andersén and László Lempert, On the group of holomorphic automorphisms of $\textbf {C}^n$, Invent. Math. 110 (1992), no. 2, 371–388. MR 1185588, DOI 10.1007/BF01231337
Shreeram S. Abhyankar and Tzuong Tsieng Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166. MR 379502
Arnaud Beauville, Brendan Hassett, Alexander Kuznetsov, and Alessandro Verra, Rationality problems in algebraic geometry, Lecture Notes in Mathematics, vol. 2172, Springer, Cham; Fondazione C.I.M.E., Florence, 2016. Lecture notes from the CIME-CIRM course held in Levico Terme, June 22–27, 2015; Edited by Rita Pardini and Gian Pietro Pirola; Fondazione CIME/CIME Foundation Subseries. MR 3587799, DOI 10.1007/978-3-319-46209-7
A. D. R. Choudary and A. Dimca, Complex hypersurfaces diffeomorphic to affine spaces, Kodai Math. J. 17 (1994), no. 2, 171–178. MR 1282208, DOI 10.2996/kmj/1138039958
Ivan Cheltsov, Adrien Dubouloz, and Jihun Park, Super-rigid affine Fano varieties, Compos. Math. 154 (2018), no. 11, 2462–2484. MR 3866510, DOI 10.1112/s0010437x18007534
C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 302652, DOI 10.2307/1970801
I. A. Chel′tsov, On a smooth four-dimensional quintic, Mat. Sb. 191 (2000), no. 9, 139–160 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 9-10, 1399–1419. MR 1805602, DOI 10.1070/SM2000v191n09ABEH000511
I. A. Chel′tsov, Birationally rigid Fano varieties, Uspekhi Mat. Nauk 60 (2005), no. 5(365), 71–160 (Russian, with Russian summary); English transl., Russian Math. Surveys 60 (2005), no. 5, 875–965. MR 2195677, DOI 10.1070/RM2005v060n05ABEH003736
Alessio Corti and Anne-Sophie Kaloghiros, The Sarkisov program for Mori fibred Calabi-Yau pairs, Algebr. Geom. 3 (2016), no. 3, 370–384. MR 3504536, DOI 10.14231/AG-2016-016
A. Clebsch, Die Geometrie auf den Flächen dritter Ordnung, J. Reine Angew. Math. 65 (1866), 359–380 (German). MR 1579329, DOI 10.1515/crll.1866.65.359
David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR 1189133, DOI 10.1007/978-1-4757-2181-2
Alessio Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), no. 2, 223–254. MR 1311348
Alessio Corti and Miles Reid (eds.), Explicit birational geometry of 3-folds, London Mathematical Society Lecture Note Series, vol. 281, Cambridge University Press, Cambridge, 2000. MR 1798978, DOI 10.1017/CBO9780511758942
Tommaso de Fernex, Erratum to: Birationally rigid hypersurfaces [ MR3049929], Invent. Math. 203 (2016), no. 2, 675–680. MR 3455160, DOI 10.1007/s00222-015-0618-4
Tommaso de Fernex, Lawrence Ein, and Mircea Mustaţă, Bounds for log canonical thresholds with applications to birational rigidity, Math. Res. Lett. 10 (2003), no. 2-3, 219–236. MR 1981899, DOI 10.4310/MRL.2003.v10.n2.a9
Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027, DOI 10.1017/CBO9781139084437
David Eisenbud and Joe Harris, 3264 and all that—a second course in algebraic geometry, Cambridge University Press, Cambridge, 2016. MR 3617981, DOI 10.1017/CBO9781139062046
Gino Fano, Sopra alcune varieta algebriche a tre dimensioni aventi tutti i generi nulli, Atti. Ac. Torino 43 (1908), 973–977.
Gino Fano, Osservazioni sopra alcune varieta non razionali aventi tutti i generi nulli, Atti. Ac. Torino 50 (1915), 1067–1072.
Gino Fano, Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del $4^\circ$ ordine, Comment. Math. Helv. 15 (1943), 71–80 (Italian). MR 10433, DOI 10.1007/BF02565634
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. MR 0476737
Edmond Halley, An easie demonstration of the analogy of the logarithmick tangents to the meridian line or sum of the secants: With various methods for computing the same to the utmost exactness, Phil. Trans. Royal Soc. 19 (1695), 202–214.
Robin Hartshorne, Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ. 26 (1986), no. 3, 375–386. MR 857224, DOI 10.1215/kjm/1250520873
Brendan Hassett, Some rational cubic fourfolds, J. Algebraic Geom. 8 (1999), no. 1, 103–114. MR 1658216
V. A. Iskovskih and Ju. I. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Mat. Sb. (N.S.) 86(128) (1971), 140–166 (Russian). MR 0291172
V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR 1668579
V. A. Iskovskih, Birational automorphisms of three-dimensional algebraic varieties, Current problems in mathematics, Vol. 12 (Russian), VINITI, Moscow, 1979, pp. 159–236, 239 (loose errata) (Russian). MR 537686
V. A. Iskovskikh, Birational rigidity of Fano hypersurfaces in the framework of Mori theory, Uspekhi Mat. Nauk 56 (2001), no. 2(338), 3–86 (Russian, with Russian summary); English transl., Russian Math. Surveys 56 (2001), no. 2, 207–291. MR 1859707, DOI 10.1070/RM2001v056n02ABEH000382
Zbigniew Jelonek, A hypersurface which has the Abhyankar-Moh property, Math. Ann. 308 (1997), no. 1, 73–84. MR 1446200, DOI 10.1007/s002080050065
János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
János Kollár, The structure of algebraic threefolds: an introduction to Mori’s program, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 2, 211–273. MR 903730, DOI 10.1090/S0273-0979-1987-15548-0
János Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36. MR 986434, DOI 10.1017/S0027763000001240
János Kollár, Flips, flops, minimal models, etc, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 113–199. MR 1144527
János Kollár, Nonrational hypersurfaces, J. Amer. Math. Soc. 8 (1995), no. 1, 241–249. MR 1273416, DOI 10.1090/S0894-0347-1995-1273416-8
János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180, DOI 10.1007/978-3-662-03276-3
János Kollár, Which are the simplest algebraic varieties?, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 4, 409–433. MR 1848255, DOI 10.1090/S0273-0979-01-00917-X
János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
J. Kollár, The structure of algebraic varieties, Proceedings of ICM, Seoul, 2014, Vol. I., Kyung Moon SA, http://www.icm2014.org/en/vod/proceedings.html, 2014, pp. 395–420.
J. Kollár, The rigidity theorem of Noether–Fano–Segre–Iskovskikh–Manin–Pukhlikov–Corti–Cheltsov–de Fernex–Ein–Mustaţă–Zhuang, arXiv:1807.00863 (2018),
János Kollár, Karen E. Smith, and Alessio Corti, Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, vol. 92, Cambridge University Press, Cambridge, 2004. MR 2062787, DOI 10.1017/CBO9780511734991
Maxim Kontsevich and Yuri Tschinkel, Specialization of birational types, arXiv:1708.05699 (2017)
Shigeru Kuroda, Recent developments in polynomial automorphisms: the solution of Nagata’s conjecture and afterwards [translation of MR3058805], Sugaku Expositions 29 (2016), no. 2, 177–201. MR 3561400
S. Lefschetz, L’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris, 1950 (French). MR 0033557
Yuchen Liu and Ziquan Zhuang, Birational superrigidity and K-stability of singular Fano complete intersections, arXiv e-prints (2018).
F. S. Macaulay, The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916 original; With an introduction by Paul Roberts. MR 1281612
T. Matsusaka and D. Mumford, Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. 86 (1964), 668–684. MR 171778, DOI 10.2307/2373030
Ugo Morin, Sulla razionalità dell’ipersuperficie cubica generale dello spazio lineare $S_5$, Rend. Sem. Mat. Univ. Padova 11 (1940), 108–112 (Italian). MR 20273
David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
Masayoshi Nagata, On automorphism group of $k[x,\,y]$, Kinokuniya Book Store Co., Ltd., Tokyo, 1972. Department of Mathematics, Kyoto University, Lectures in Mathematics, No. 5. MR 0337962
Max Noether, Ueber Flächen, welche Schaaren rationaler Curven besitzen, Math. Ann. 3 (1870), no. 2, 161–227 (German). MR 1509694, DOI 10.1007/BF01443982
M. Noether, Zur Grundlegung der Theorie der algebraischen Raumcurven, J. Reine Angew. Math. 93 (1882), 271–318 (German). MR 1579926, DOI 10.1515/crll.1882.93.271
Johannes Nicaise and Evgeny Shinder, The motivic nearby fiber and degeneration of stable rationality, arXiv:1708.027901 (2017).
Keiji Oguiso, Isomorphic quartic K3 surfaces in the view of Cremona and projective transformations, Taiwanese J. Math. 21 (2017), no. 3, 671–688. MR 3661387, DOI 10.11650/tjm/7833
A. V. Pukhlikov, Birational isomorphisms of four-dimensional quintics, Invent. Math. 87 (1987), no. 2, 303–329. MR 870730, DOI 10.1007/BF01389417
A. V. Pukhlikov, A remark on the theorem of V. A. Iskovskikh and Yu. I. Manin on a three-dimensional quartic, Trudy Mat. Inst. Steklov. 208 (1995), no. Teor. Chisel, Algebra i Algebr. Geom., 278–289 (Russian). Dedicated to Academician Igor′Rostislavovich Shafarevich on the occasion of his seventieth birthday (Russian). MR 1730270
Aleksandr V. Pukhlikov, Birational automorphisms of Fano hypersurfaces, Invent. Math. 134 (1998), no. 2, 401–426. MR 1650332, DOI 10.1007/s002220050269
A. V. Pukhlikov, Birationally rigid Fano hypersurfaces, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 6, 159–186 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 6, 1243–1269. MR 1970356, DOI 10.1070/IM2002v066n06ABEH000413
Aleksandr Pukhlikov, Birationally rigid varieties, Mathematical Surveys and Monographs, vol. 190, American Mathematical Society, Providence, RI, 2013. MR 3060242, DOI 10.1090/surv/190
Miles Reid, Undergraduate algebraic geometry, London Mathematical Society Student Texts, vol. 12, Cambridge University Press, Cambridge, 1988. MR 982494, DOI 10.1017/CBO9781139163699
Miles Reid, Chapters on algebraic surfaces, Complex algebraic geometry (Park City, UT, 1993) IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence, RI, 1997, pp. 3–159. MR 1442522, DOI 10.1090/pcms/003/02
F. Russo and G. Staglianò, Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds, arXiv e-prints (2017).
George Salmon, A treatise on the analytic geometry of three dimensions. Vol. II, 5th ed., Chelsea Publishing Co., New York, 1965. Edited by Reginald A. P. Rogers. MR 0200123
V. G. Sarkisov, Birational automorphisms of conic bundles, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 918–945, 974 (Russian). MR 587343
S. Schreieder, Stably irrational hypersurfaces of small slopes, arXiv e-prints (2018).
B. Segre, A note on arithmetical properties of cubic surfaces, J. London Math. Soc 18 (1943), 24–31. MR 0009471, DOI 10.1112/jlms/s1-18.1.24
I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917, DOI 10.1007/978-3-642-96200-4
C. L. Siegel, Topics in complex function theory. Vol. I: Elliptic functions and uniformization theory, Interscience Tracts in Pure and Applied Mathematics, No. 25, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1969. Translated from the original German by A. Shenitzer and D. Solitar. MR 0257326
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Elias M. Stein and Rami Shakarchi, Complex analysis, Princeton Lectures in Analysis, vol. 2, Princeton University Press, Princeton, NJ, 2003. MR 1976398
Ichiro Shimada and Tetsuji Shioda, On a smooth quartic surface containing 56 lines which is isomorphic as a $K3$ surface to the Fermat quartic, Manuscripta Math. 153 (2017), no. 1-2, 279–297. MR 3635983, DOI 10.1007/s00229-016-0886-3
Ivan P. Shestakov and Ualbai U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc. 17 (2004), no. 1, 197–227. MR 2015334, DOI 10.1090/S0894-0347-03-00440-5
Masakazu Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace $\textbf {C}^{2}$, J. Math. Soc. Japan 26 (1974), 241–257 (French). MR 338423, DOI 10.2969/jmsj/02620241
Charlie Stibitz and Ziquan Zhuang, K-stability of birationally superrigid Fano varieties, ArXiv e-prints (2018).
Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. MR 1790619, DOI 10.1007/978-3-0348-8440-2
Claire Voisin, Unirational threefolds with no universal codimension $2$ cycle, Invent. Math. 201 (2015), no. 1, 207–237. MR 3359052, DOI 10.1007/s00222-014-0551-y
Ziquan Zhuang, Birational superrigidity and K-stability of Fano complete intersections of index one (with an appendix written jointly with Charlie Stibitz), arXiv e-prints (2018).