×

Algebraic hypersurfaces. (English) Zbl 1470.14085

This article is a very enjoyable and instructive introduction to Birational Geometry.
By focusing on algebraic hypersurfaces, the author presents a very accessible survey of the main questions, ideas, tools and techniques used in that field. The article stays clear of technical terms and expert terminology. After defining (affine and projective) hypersurfaces and (bi-)rational maps, the author asks the question driving of this exposition: when are hypersurfaces the same (isomorphic) or close to being the same (birational)? In particular, which hypersurfaces are close to projective space (rational), or on the contrary close to no other hypersurface (rigid)?
The article surveys results on rational and rigid hypersurfaces, and sketches the connection with the classification of varieties developed by Enriques, Iitaka and Mori. The last section presents some open questions.

MSC:

14J70 Hypersurfaces and algebraic geometry
14M20 Rational and unirational varieties
14E08 Rationality questions in algebraic geometry
14N25 Varieties of low degree
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abhyankar, S. S., Lectures on expansion techniques in algebraic geometry, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 57, iv+168 pp. (1977), Tata Institute of Fundamental Research, Bombay · Zbl 0818.14001
[2] Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J., Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 267, xvi+386 pp. (1985), Springer-Verlag, New York · Zbl 0559.14017 · doi:10.1007/978-1-4757-5323-3
[3] Araujo, Carolina; Koll\'{a}r, J\'{a}nos, Rational curves on varieties. Higher dimensional varieties and rational points, Budapest, 2001, Bolyai Soc. Math. Stud. 12, 13-68 (2003), Springer, Berlin · Zbl 1080.14521 · doi:10.1007/978-3-662-05123-8\_3
[4] Anders\'{e}n, Erik; Lempert, L\'{a}szl\'{o}, On the group of holomorphic automorphisms of \({\bf C}^n\), Invent. Math., 110, 2, 371-388 (1992) · Zbl 0770.32015 · doi:10.1007/BF01231337
[5] Abhyankar, Shreeram S.; Moh, Tzuong Tsieng, Embeddings of the line in the plane, J. Reine Angew. Math., 276, 148-166 (1975) · Zbl 0332.14004
[6] Beauville, Arnaud; Hassett, Brendan; Kuznetsov, Alexander; Verra, Alessandro, Rationality problems in algebraic geometry, Lecture Notes in Mathematics 2172, viii+167 pp. (2016), Springer, Cham; Fondazione C.I.M.E., Florence · Zbl 1364.14001 · doi:10.1007/978-3-319-46209-7
[7] Choudary, A. D. R.; Dimca, A., Complex hypersurfaces diffeomorphic to affine spaces, Kodai Math. J., 17, 2, 171-178 (1994) · Zbl 0820.14030 · doi:10.2996/kmj/1138039958
[8] Cheltsov, Ivan; Dubouloz, Adrien; Park, Jihun, Super-rigid affine Fano varieties, Compos. Math., 154, 11, 2462-2484 (2018) · Zbl 1408.14052 · doi:10.1112/s0010437x18007534
[9] Clemens, C. Herbert; Griffiths, Phillip A., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2), 95, 281-356 (1972) · Zbl 0214.48302 · doi:10.2307/1970801
[10] Chel\cprime tsov, I. A., On a smooth four-dimensional quintic, Mat. Sb.. Sb. Math., 191 191, 9-10, 1399-1419 (2000) · Zbl 0998.14018 · doi:10.1070\allowbreak/SM2000v191n09ABEH000511
[11] Chel\cprime tsov, I. A., Birationally rigid Fano varieties, Uspekhi Mat. Nauk. Russian Math. Surveys, 60 60, 5, 875-965 (2005) · Zbl 1145.14032 · doi:10.1070\allowbreak/RM2005v060n05ABEH003736
[12] Corti, Alessio; Kaloghiros, Anne-Sophie, The Sarkisov program for Mori fibred Calabi-Yau pairs, Algebr. Geom., 3, 3, 370-384 (2016) · Zbl 1437.14024 · doi:10.14231/AG-2016-016
[13] Clebsch, A., Die Geometrie auf den Fl\"{a}chen dritter Ordnung, J. Reine Angew. Math., 65, 359-380 (1866) · ERAM 065.1709cj · doi:10.1515/crll.1866.65.359
[14] Cox, David; Little, John; O’Shea, Donal, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, xii+513 pp. (1992), Springer-Verlag, New York · Zbl 0756.13017 · doi:10.1007/978-1-4757-2181-2
[15] Corti, Alessio, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom., 4, 2, 223-254 (1995) · Zbl 0866.14007
[16] Explicit birational geometry of 3-folds, London Mathematical Society Lecture Note Series 281, vi+349 pp. (2000), Cambridge University Press, Cambridge · Zbl 0942.00009 · doi:10.1017/CBO9780511758942
[17] de Fernex, Tommaso, Erratum to: Birationally rigid hypersurfaces [MR3049929], Invent. Math., 203, 2, 675-680 (2016) · Zbl 1441.14045 · doi:10.1007/s00222-015-0618-4
[18] de Fernex, Tommaso; Ein, Lawrence; Musta\c{t}\u{a}, Mircea, Bounds for log canonical thresholds with applications to birational rigidity, Math. Res. Lett., 10, 2-3, 219-236 (2003) · Zbl 1067.14013 · doi:10.4310/MRL.2003.v10.n2.a9
[19] Dolgachev, Igor V., Classical algebraic geometry, xii+639 pp. (2012), Cambridge University Press, Cambridge · Zbl 1252.14001 · doi:10.1017/CBO9781139084437
[20] David Eisenbud and Joe Harris, 3264 and all that: A second course in algebraic geometry, Cambridge University Press, Cambridge, 2016.. · Zbl 1341.14001
[21] Gino Fano, Sopra alcune varieta algebriche a tre dimensioni aventi tutti i generi nulli, Atti. Ac. Torino 43 (1908), 973-977. · JFM 39.0718.03
[22] Gino Fano, Osservazioni sopra alcune varieta non razionali aventi tutti i generi nulli, Atti. Ac. Torino 50 (1915), 1067-1072. · JFM 45.1357.03
[23] Fano, Gino, Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del \(4^\circ\) ordine, Comment. Math. Helv., 15, 71-80 (1943) · JFM 68.0400.02 · doi:10.1007/BF02565634
[24] Griffiths, Phillip; Harris, Joseph, Principles of algebraic geometry, xii+813 pp. (1978), Wiley-Interscience [John Wiley & Sons], New York · Zbl 0836.14001
[25] Grothendieck, Alexander, Cohomologie locale des faisceaux coh\'{e}rents et th\'{e}or\`“emes de Lefschetz locaux et globaux \((SGA 2)\), vii+287 pp. (1968), North-Holland Publishing Co., Amsterdam; Masson & Cie, \'”{E}diteur, Paris · Zbl 0197.47202
[26] 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.
[27] Hartshorne, Robin, Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ., 26, 3, 375-386 (1986) · Zbl 0613.14008 · doi:10.1215/kjm/1250520873
[28] Hassett, Brendan, Some rational cubic fourfolds, J. Algebraic Geom., 8, 1, 103-114 (1999) · Zbl 0961.14029
[29] Iskovskih, V. A.; Manin, Ju. I., Three-dimensional quartics and counterexamples to the L\"{u}roth problem, Mat. Sb. (N.S.), 86(128), 140-166 (1971) · Zbl 0222.14009
[30] Iskovskikh, V. A.; Prokhorov, Yu. G., Fano varieties. Algebraic geometry, V, Encyclopaedia Math. Sci. 47, 1-247 (1999), Springer, Berlin · Zbl 0912.14013
[31] Iskovskih, V. A., Birational automorphisms of three-dimensional algebraic varieties. Current problems in mathematics, Vol. 12 (Russian), 159-236, 239 (loose errata) (1979), VINITI, Moscow · Zbl 0415.14025
[32] Iskovskikh, V. A., Birational rigidity of Fano hypersurfaces in the framework of Mori theory, Uspekhi Mat. Nauk. Russian Math. Surveys, 56 56, 2, 207-291 (2001) · Zbl 0991.14010 · doi:10.1070/RM2001v056n02ABEH000382
[33] Jelonek, Zbigniew, A hypersurface which has the Abhyankar-Moh property, Math. Ann., 308, 1, 73-84 (1997) · Zbl 0870.14004 · doi:10.1007/s002080050065
[34] Koll\'{a}r, J\'{a}nos; Mori, Shigefumi, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, viii+254 pp. (1998), Cambridge University Press, Cambridge · Zbl 0926.14003 · doi:10.1017/CBO9780511662560
[35] Koll\'{a}r, J\'{a}nos, The structure of algebraic threefolds: an introduction to Mori’s program, Bull. Amer. Math. Soc. (N.S.), 17, 2, 211-273 (1987) · Zbl 0649.14022 · doi:10.1090/S0273-0979-1987-15548-0
[36] Koll\'{a}r, J\'{a}nos, Flops, Nagoya Math. J., 113, 15-36 (1989) · Zbl 0645.14004 · doi:10.1017\allowbreak/S0027763000001240
[37] J. Koll\'ar, Flips, flops, minimal models, etc, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 113-199. · Zbl 0755.14003
[38] Koll\'{a}r, J\'{a}nos, Nonrational hypersurfaces, J. Amer. Math. Soc., 8, 1, 241-249 (1995) · Zbl 0839.14031 · doi:10.2307/2152888
[39] Koll\'{a}r, J\'{a}nos, 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] 32, viii+320 pp. (1996), Springer-Verlag, Berlin · Zbl 0877.14012 · doi:10.1007/978-3-662-03276-3
[40] Koll\'{a}r, J\'{a}nos, Which are the simplest algebraic varieties?, Bull. Amer. Math. Soc. (N.S.), 38, 4, 409-433 (2001) · Zbl 0978.14039 · doi:10.1090/S0273-0979-01-00917-X
[41] Koll\'{a}r, J\'{a}nos, Singularities of the minimal model program, Cambridge Tracts in Mathematics 200, x+370 pp. (2013), Cambridge University Press, Cambridge · Zbl 1282.14028 · doi:10.1017/CBO9781139547895
[42] J. Koll\'ar, 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. · Zbl 1373.14001
[43] J. Koll\'ar, The rigidity theorem of Noether-Fano-Segre-Iskovskikh-Manin-Pukhlikov-Corti-Cheltsov-deFernex-Ein-Mustata-Zhuang, arXiv:1807.00863 (2018), · Zbl 1473.14080
[44] Koll\'{a}r, J\'{a}nos; Smith, Karen E.; Corti, Alessio, Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics 92, vi+235 pp. (2004), Cambridge University Press, Cambridge · Zbl 1060.14073 · doi:10.1017/CBO9780511734991
[45] Maxim Kontsevich and Yuri Tschinkel, Specialization of birational types, arXiv:1708.05699 (2017) · Zbl 1420.14030
[46] Kuroda, Shigeru, Recent developments in polynomial automorphisms: the solution of Nagata’s conjecture and afterwards [translation of MR3058805], Sugaku Expositions, 29, 2, 177-201 (2016)
[47] Lefschetz, S., L’analysis situs et la g\'{e}om\'{e}trie alg\'{e}brique, vi+154 pp. (1950), Gauthier-Villars, Paris · Zbl 0035.10202
[48] Yuchen Liu and Ziquan Zhuang, Birational superrigidity and K-stability of singular Fano complete intersections, arXiv e-prints (2018). · Zbl 1391.14081
[49] Macaulay, F. S., The algebraic theory of modular systems, Cambridge Mathematical Library, xxxii+112 pp. (1994), Cambridge University Press, Cambridge · Zbl 0802.13001
[50] Matsusaka, T.; Mumford, D., Two fundamental theorems on deformations of polarized varieties, Amer. J. Math., 86, 668-684 (1964) · Zbl 0128.15505 · doi:10.2307/2373030
[51] Morin, Ugo, Sulla razionalit\`“a dell”ipersuperficie cubica generale dello spazio lineare \(S_5\), Rend. Sem. Mat. Univ. Padova, 11, 108-112 (1940) · Zbl 0024.17301
[52] Mumford, David, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, x+186 pp. (1976), Springer-Verlag, Berlin-New York · Zbl 0821.14001
[53] Nagata, Masayoshi, On automorphism group of \(k[x,\,y]\), v+53 pp. (1972), Kinokuniya Book-Store Co., Ltd., Tokyo · Zbl 0306.14001
[54] Noether, Max, Ueber Fl\"{a}chen, welche Schaaren rationaler Curven besitzen, Math. Ann., 3, 2, 161-227 (1870) · JFM 02.0616.02 · doi:10.1007/BF01443982
[55] Noether, M., Zur Grundlegung der Theorie der algebraischen Raumcurven, J. Reine Angew. Math., 93, 271-318 (1882) · JFM 14.0669.03 · doi:10.1515/crll.1882.93.271
[56] Johannes Nicaise and Evgeny Shinder, The motivic nearby fiber and degeneration of stable rationality, arXiv:1708.027901 (2017). · Zbl 1455.14029
[57] Oguiso, Keiji, Isomorphic quartic K3 surfaces in the view of Cremona and projective transformations, Taiwanese J. Math., 21, 3, 671-688 (2017) · Zbl 1391.14076 · doi:10.11650/tjm/7833
[58] Pukhlikov, A. V., Birational isomorphisms of four-dimensional quintics, Invent. Math., 87, 2, 303-329 (1987) · Zbl 0613.14011 · doi:10.1007/BF01389417
[59] 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, Dedicated to Academician Igor Rostislavovich Shafarevich on the occasion of his seventieth birthday (Russian). · Zbl 0880.14020
[60] Pukhlikov, Aleksandr V., Birational automorphisms of Fano hypersurfaces, Invent. Math., 134, 2, 401-426 (1998) · Zbl 0964.14011 · doi:10.1007/s002220050269
[61] Pukhlikov, A. V., Birationally rigid Fano hypersurfaces, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Math., 66 66, 6, 1243-1269 (2002) · Zbl 1083.14012 · doi:10.1070/IM2002v066n06ABEH000413
[62] Pukhlikov, Aleksandr, Birationally rigid varieties, Mathematical Surveys and Monographs 190, vi+365 pp. (2013), American Mathematical Society, Providence, RI · Zbl 1297.14001 · doi:10.1090/surv/190
[63] Reid, Miles, Undergraduate algebraic geometry, London Mathematical Society Student Texts 12, viii+129 pp. (1988), Cambridge University Press, Cambridge · Zbl 0701.14001 · doi:10.1017/CBO9781139163699
[64] Reid, Miles, Chapters on algebraic surfaces. Complex algebraic geometry, Park City, UT, 1993, IAS/Park City Math. Ser. 3, 3-159 (1997), Amer. Math. Soc., Providence, RI · Zbl 0910.14016
[65] F. Russo and G. Staglian\`o, Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds, arXiv e-prints (2017). · Zbl 1442.14051
[66] Salmon, George, A treatise on the analytic geometry of three dimensions. Vol. II, Fifth edition. Edited by Reginald A. P. Rogers, xvi+334 pp. (1965), Chelsea Publishing Co., New York
[67] Sarkisov, V. G., Birational automorphisms of conic bundles, Izv. Akad. Nauk SSSR Ser. Mat., 44, 4, 918-945, 974 (1980) · Zbl 0453.14017
[68] S. Schreieder, Stably irrational hypersurfaces of small slopes, arXiv e-prints (2018). · Zbl 1442.14138
[69] Segre, B., A note on arithmetical properties of cubic surfaces, J. London Math. Soc, 18, 24-31 (1943) · Zbl 0060.09205 · doi:10.1112/jlms/s1-18.1.24
[70] Shafarevich, I. R., Basic algebraic geometry, xv+439 pp. (1974), Springer-Verlag, New York-Heidelberg · Zbl 0284.14001
[71] Siegel, C. L., Topics in complex function theory. Vol. I: Elliptic functions and uniformization theory, Translated from the original German by A. Shenitzer and D. Solitar. Interscience Tracts in Pure and Applied Mathematics, No. 25, ix+186 pp. (1969), Wiley-Interscience A Division of John Wiley & Sons, New York-London-Sydney · Zbl 0635.30002
[72] Silverman, Joseph H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, xii+400 pp. (1986), Springer-Verlag, New York · Zbl 0585.14026 · doi:10.1007/978-1-4757-1920-8
[73] Stein, Elias M.; Shakarchi, Rami, Complex analysis, Princeton Lectures in Analysis 2, xviii+379 pp. (2003), Princeton University Press, Princeton, NJ · Zbl 1020.30001
[74] Shimada, Ichiro; Shioda, Tetsuji, On a smooth quartic surface containing 56 lines which is isomorphic as a \(K3\) surface to the Fermat quartic, Manuscripta Math., 153, 1-2, 279-297 (2017) · Zbl 1387.14106 · doi:10.1007/s00229-016-0886-3
[75] Shestakov, Ivan P.; Umirbaev, Ualbai U., The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc., 17, 1, 197-227 (2004) · Zbl 1056.14085 · doi:10.1090/S0894-0347-03-00440-5
[76] Suzuki, Masakazu, Propri\'{e}t\'{e}s topologiques des polyn\^omes de deux variables complexes, et automorphismes alg\'{e}briques de l’espace \({\bf C}^2\), J. Math. Soc. Japan, 26, 241-257 (1974) · Zbl 0276.14001 · doi:10.2969/jmsj/02620241
[77] Charlie Stibitz and Ziquan Zhuang, K-stability of birationally superrigid Fano varieties, ArXiv e-prints (2018). · Zbl 1425.14035
[78] van den Essen, Arno, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics 190, xviii+329 pp. (2000), Birkh\"{a}user Verlag, Basel · Zbl 0962.14037 · doi:10.1007/978-3-0348-8440-2
[79] Voisin, Claire, Unirational threefolds with no universal codimension \(2\) cycle, Invent. Math., 201, 1, 207-237 (2015) · Zbl 1327.14223 · doi:10.1007/s00222-014-0551-y
[80] 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). · Zbl 1456.14050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.