Hyperelliptic and trigonal Fano threefolds

V. V. Przyjalkowski ; I. A. Cheltsov ; K. A. Shramov

Izvestiya. Mathematics , Tome 69 (2005) no. 2, pp. 365-421.

Résumé

We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein singularities which are not intersections of quadrics. We also study rationality questions for most of these varieties.

Bibliographie

[1] Arnold V. I., Gusein-Zade S. M., Varchenko A. N., Osobennosti differentsiruemykh otobrazhenii. Klassifikatsiya kriticheskikh tochek, kaustik, volnovykh frontov, Nauka, M., 1982 | MR

[2] Gizatulin M., “Ratsionalnye $G$-poverkhnosti”, Izv. AN SSSR. Ser. matem., 44:1 (1980), 110–144 | MR

[3] Grinenko M. M., “Biratsionalnye avtomorfizmy trekhmernogo dvoinogo konusa”, Matem. sb., 189:7 (1998), 37–52 | MR | Zbl

[4] Grinenko M. M., “Biratsionalnye avtomorfizmy trekhmernoi dvoinoi kvadriki s prosteishei osobennostyu”, Matem. sb., 189:1 (1998), 101–118 | MR | Zbl

[5] Grinenko M. M., “Biratsionalnye svoistva puchkov poverkhnostei del Petstso stepeni 1 i 2”, Matem. sb., 191:5 (2000), 17–38 | MR | Zbl

[6] Grinenko M. M., “O dvoinom konuse nad poverkhnostyu Veroneze”, Izv. RAN. Ser. matem., 67:3 (2003), 5–22 | MR | Zbl

[7] Grinenko M. M., “Struktury Mori trekhmernogo mnogoobraziya Fano indeksa 2 i stepeni 1”, Tr. MIRAN, 246 (2004), 103–128 | MR | Zbl

[8] Iskovskikh V. A., “O biratsionalnykh formakh ratsionalnykh poverkhnostei”, Izv. AN SSSR. Ser. matem., 29:6 (1965), 1417–1433 | Zbl

[9] Iskovskikh V. A., “Ratsionalnye poverkhnosti s puchkom ratsionalnykh krivykh”, Matem. sb., 74:4 (1967), 608–638 | Zbl

[10] Iskovskikh V. A., “Ratsionalnye poverkhnosti s puchkom ratsionalnykh krivykh i polozhitelnym kvadratom kanonicheskogo klassa”, Matem. sb., 83:1 (1970), 90–119 | Zbl

[11] Iskovskikh V. A., “Biratsionalnye svoistva poverkhnosti stepeni 4 v $\mathbb P_k^4$”, Matem. sb., 88:1 (1972), 31–37 | Zbl

[12] Iskovskikh V. A., “Trekhmernye mnogoobraziya Fano, I”, Izv. AN SSSR. Ser. matem., 41:3 (1977), 516–562 | MR | Zbl

[13] Iskovskikh V. A., “Trekhmernye mnogoobraziya Fano, II”, Izv. AN SSSR. Ser. matem., 42:3 (1978), 504–549 | MR

[14] Iskovskikh V. A., “Minimalnye modeli algebraicheskikh poverkhnostei nad sovershennymi polyami”, Izv. AN SSSR. Ser. matem., 43:1 (1979), 19–43 | MR | Zbl

[15] Iskovskikh V. A., “Antikanonicheskie modeli trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhniki. Sovr. probl. matem., 12, VINITI, M., 1979, 59–157 | MR

[16] Iskovskikh V. A., “Biratsionalnye avtomorfizmy trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhniki. Sovr. probl. matem., 12, VINITI, M., 1979, 159–236 | MR

[17] Iskovskikh V. A., Lektsii o trekhmernykh algebraicheskikh mnogoobraziyakh. Mnogoobraziya Fano, Izd-vo MGU, M., 1988 | Zbl

[18] Iskovskikh V. A., “K probleme ratsionalnosti rassloenii na koniki”, Matem. sb., 182:1 (1991), 114–121 | MR

[19] Iskovskikh V. A., “O kriterii ratsionalnosti dlya rassloenii na koniki”, Matem. sb., 187:7 (1996), 75–92 | MR | Zbl

[20] Iskovskikh V. A., “Faktorizatsiya biratsionalnykh otobrazhenii ratsionalnykh poverkhnostei s tochki zreniya teorii Mori”, UMN, 51:4 (1996), 3–72 | MR | Zbl

[21] Iskovskikh V. A., “O probleme ratsionalnosti dlya trekhmernykh algebraicheskikh mnogoobrazii”, Tr. MIRAN, 218 (1997), 190–232 | MR | Zbl

[22] Iskovskikh V. A., Manin Yu. I., “Trekhmernye kvartiki i kontrprimery k probleme Lyurota”, Matem. sb., 86:1 (1971), 140–166 | Zbl

[23] Manin Yu. I., “Ratsionalnye poverkhnosti nad sovershennymi polyami”, Publ. Math. IHES, 30 (1966), 55–113 | MR

[24] Manin Yu. I., “Ratsionalnye poverkhnosti nad sovershennymi polyami, II”, Matem. sb., 72:2 (1967), 161–192 | MR | Zbl

[25] Manin Yu. I., Kubicheskie formy, Nauka, M., 1972 | MR

[26] Manin Yu. I., Tsfasman M. A., “Ratsionalnye mnogoobraziya: algebra, geometriya, arifmetika”, UMN, 41:2 (1986), 43–94 | MR | Zbl

[27] Pukhlikov A. V., “Biratsionalnye avtomorfizmy dvoinogo prostranstva i dvoinoi kvadriki”, Izv. AN SSSR. Ser. matem., 52:1 (1988), 229–239 | MR

[28] Pukhlikov A. V., “Biratsionalnye avtomorfizmy trekhmernoi kvartiki s prosteishei osobennostyu”, Matem. sb., 135(177):4 (1988), 472–496 | Zbl

[29] Pukhlikov A. V., “Maksimalnye osobennosti na mnogoobrazii Fano $V^3_6$”, Vestn. MGU. Ser. 1. Matem., mekhanika, 1989, no. 2, 47–50 | MR | Zbl

[30] Pukhlikov A. V., “Biratsionalnye avtomorfizmy trekhmernykh algebraicheskikh mnogoobrazii s puchkom poverkhnostei del Petstso”, Izv. RAN. Ser. matem., 62:1 (1998), 123–164 | MR | Zbl

[31] Tikhomirov A. S., “Geometriya poverkhnosti Fano dvoinogo prostranstva $\mathbb P^3$ s vetvleniem v kvartike”, Izv. AN SSSR. Ser. matem., 44:2 (1980), 415–442 | MR | Zbl

[32] Tikhomirov A. S., “Srednii yakobian dvoinogo prostranstva $\mathbb P^3$, razvetvlennogo v kvartike”, Izv. AN SSSR. Ser. matem., 44:6 (1980), 1329–1377 | MR | Zbl

[33] Tikhomirov A. S., “Osobennosti teta-difizora srednego yakobiana dvoinogo prostranstva $\mathbb P^3$ indeksa 2”, Izv. AN SSSR. Ser. matem., 46:5 (1982), 1062–1081 | MR

[34] Tikhomirov A. S., “Otobrazhenie Abelya–Yakobi sekstiki roda tri na dvoinoe nakrytie $\mathbb P^3$ indeksa 2”, DAN SSSR, 286:4 (1986), 821–824 | MR | Zbl

[35] Tsfasman M. A., “Arifmetika osobykh poverkhnostei del Petstso”, UMN, 38:6 (1983), 131–132 | MR | Zbl

[36] Tyurin A. N., “O peresechenii kvadrik”, UMN, 30:6 (1975), 51–99 | MR | Zbl

[37] Tyurin A. N., “Promezhutochnyi yakobian trekhmernykh mnogoobrazii”, Itogi nauki i tekhniki. Sovr. probl. matem., 12, VINITI, M., 1979, 5–57 | MR

[38] Khashin S., “Biratsionalnye avtomorfizmy trekhmernogo dvoinogo konusa nad poverkhnostyu Veroneze”, Vestn. MGU. Ser. 1. Matem., mekhanika, 1984, no. 1, 13–16 | MR | Zbl

[39] Cheltsov I. A., “Osobennosti trekhmernykh mnogoobrazii, obladayuschikh obilnym effektivnym divizorom – gladkoi poverkhnostyu kodairovoi razmernosti nul”, Matem. zametki, 59:4 (1996), 618–626 | MR | Zbl

[40] Cheltsov I. A., “Trekhmernye mnogoobraziya, obladayuschie divizorom s chislenno trivialnym kanonicheskim klassom”, UMN, 51:1 (1996), 177–178 | MR

[41] Cheltsov I. A., “Ogranichennost trekhmernykh mnogoobrazii Fano tselogo indeksa”, Matem. zametki, 66:3 (1999), 445–451 | MR

[42] Cheltsov I. A., “Dvoinoe prostranstvo s dvoinoi pryamoi”, Matem. sb., 195:10 (2004), 109–156 | MR

[43] Cheltsov I. A., “Metod vyrozhdeniya i neratsionalnost trekhmernykh mnogoobrazii s puchkom poverkhnostei del Petstso”, UMN, 59:4 (2004), 203–204 | MR

[44] Shokurov V. V., “Teorema Netera–Enrikvesa o kanonicheskikh krivykh”, Matem. sb., 86:3 (1971), 367–408 | MR | Zbl

[45] Shokurov V. V., “Gladkost obschego antikanonicheskogo divizora na trekhmernom mnogoobrazii Fano”, Izv. AN SSSR. Ser. matem., 43:2 (1979), 430–441 | MR | Zbl

[46] Shokurov V. V., “Suschestvovanie pryamoi na mnogoobraziyakh Fano”, Izv. AN SSSR. Ser. matem., 43:4 (1979), 922–964 | MR | Zbl

[47] Shokurov V. V., “Mnogoobraziya Prima: teoriya i prilozheniya”, Izv. AN SSSR. Ser. matem., 47:4 (1983), 785–855 | MR | Zbl

[48] Alexeev V., “Theorems about good divisors on log Fano varieties (case of index $r>n-2$)”, Lect. Note Math., 1479, 1991, 1–9 | MR | Zbl

[49] Ambro F., “Ladders on Fano varieties”, J. Math. Sci. (New York), 94 (1999), 1126–1135 | DOI | MR | Zbl

[50] Andreotti A., Frankel T., “The Lefschetz theorem on hyperplane sections”, Ann. Math., 69 (1959), 713–717 | DOI | MR | Zbl

[51] Artin M., “Some numerical criteria of contractability of curves on algebraic surfaces”, Amer. J. Math., 84 (1962), 485–496 | DOI | MR | Zbl

[52] Bardelli F., “Polarized mixed Hodge structures: on irrationality of threefolds via degeneration”, Ann. Mat. Pura et Appl., 137 (1984), 287–369 | DOI | MR | Zbl

[53] Barth W., Peters C., van de Ven A., Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin–New York, 1984 | MR | Zbl

[54] Beauville A., “Varietes de Prym et jacobiennes intermediaires”, Ann. Sci. Ecole Norm. Sup., 10 (1977), 309–391 | MR | Zbl

[55] Bott R., “On a theorem of Lefschetz”, Mich. Math. J., 6 (1959), 211–216 | DOI | MR | Zbl

[56] Brown G., Corti A., Zucconi F., “Birational geometry of 3-fold Mori fibre spaces”, Proceedings the Fano conference, eds. Alberto Collino, Alberto Conte, Marina Marchisio, Dipartimento di Matematica dell'università di Torino, 2004 ; arXiv: math.AG/0307301 | MR

[57] Campana F., Flenner H., “Projective threefolds containing a smooth rational surface with ample normal bundle”, J. Reine Angew. Math., 440 (1993), 77–98 | MR | Zbl

[58] Cheltsov I., “Non-rational nodal quartic threefolds”, Pacific J. Math., 2005 (to appear); arXiv: math.AG/0405150

[59] Cheltsov I., Park J., Sextic double solids, arXiv: math.AG/0404452 | MR

[60] Clemens H., “Double solids”, Adv. in Math., 47 (1983), 107–230 | DOI | MR | Zbl

[61] Clemens H., “The quartic double solid revisited”, Complex geometry and Lie theory (Sundance, UT, 1989), Proc. Sympos. Pure Math., 53, 1991, 89–101 | MR | Zbl

[62] Clemens H., Griffiths P., “The intermediate Jacobian of the cubic threefold”, Ann. of Math., 95 (1972), 73–100 | DOI | MR

[63] Corti A., “Singularities of linear systems and 3-fold birational geometry”, L. M. S. Lecture Note Series, 281, 2000, 259–312 | MR | Zbl

[64] Corti A., Mella M., “Birational geometry of terminal quartic 3-folds, I”, Amer. J. Math., 126:4 (2004), 739–761 | DOI | MR | Zbl

[65] Cynk S., “Defect of a nodal hypersurface”, Manuscripta Math., 104 (2001), 325–331 | DOI | MR | Zbl

[66] Cynk S., “Cohomologies of a double covering of a non-singular algebraic 3-fold”, Math. Z., 240 (2002), 731–743 | DOI | MR | Zbl

[67] Demazure M., “Surfaces de del Pezzo”, Lecture Notes in Math., 777, Springer, Berlin–Heidelberg–New York, 1989, 21–69

[68] Dolgachev I., “Weighted projective varieties”, Lect. Notes Math., 956, 1982, 34–71 | MR | Zbl

[69] Eisenbud D., Harris J., “On varieties of minimal degree”, Proc. Symp. Pure Math., 46 (1987), 3–13 | MR | Zbl

[70] Elkik R., “Rationalite des singularites canoniques”, Invent. Math., 64 (1981), 1–6 | DOI | MR | Zbl

[71] Enriques F., “Sepra una involuzione non razionale de dello spacio”, Rend. Acc. Lincei, 21 (1912), 81–83 | Zbl

[72] Fano G., “Sulle varietá algebriche a tre dimensioni aventi tutti i generi nulli”, Atti Congr. Int. Bologna, V. IV, 1929, 115–121

[73] Fano G., “Su alcune varietá algebriche a tre dimensioni a curve sezioni canoniche”, Scritti Mat. offerti a L. Berzolari Ist. Mat. R. Univ. Pavia, 1936, 329–349 | Zbl

[74] Fano G., “Sulle varietá algebriche a tre dimensioni a curve sezioni canoniche”, Mem. Accad. d'Italiana, VIII (1937), 23–64

[75] Fano G., “Su alcune varietá algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche”, Comment. Math. Helv., 14 (1942), 202–211 | DOI | MR | Zbl

[76] Iano-Fletcher A. R., “Working with weighted complete intersections”, L. M. S. Lecture Note Series, 281, 2000, 101–173 | MR

[77] Fujita T., “On the structure of polarized varieties with $\Delta$-genera zero”, J. Fac. Sci. Univ. Tokyo Sec. IA, 22 (1975), 103–115 | MR | Zbl

[78] Fujita T., “On singular Del Pezzo varieties”, Lecture Notes in Math., 1417, 1990, 117–128 | MR | Zbl

[79] Fujita T., Classification theories of polarized varieties, L. M. S. Lecture Note Series, 155, 1990 | MR | Zbl

[80] Furushima M., “Mukai–Umemura's example of a Fano 3-fold with genus 12 as a compactification of $\mathbb C^3$”, Nagoya Math. J., 127 (1990), 145–165 | MR

[81] Griffiths P., Harris J., Principles of algebraic geometry, John Wiley, New York, 1978 | MR | Zbl

[82] Grinenko M., “On the birational rigidity of some pencils of del Pezzo surfaces”, J. Math. Sciences, 102 (2000), 3933–3937 | DOI | MR | Zbl

[83] Hartshorne R., Algebraic geometry, Graduare Text in Mathematics, 52, Springer-Verlag, New York–Heidelberg–Berlin, 1977 | MR | Zbl

[84] Jahnke P., Radloff I., Gorenstein Fano threefolds with base points in the anticanonical system, arXiv: math.AG/0404156 | MR

[85] Ishii S., “Quasi-Gorenstein Fano 3-folds with isolated non-rational loci”, Comp. Math., 77 (1991), 335–341 | MR | Zbl

[86] Iskovskikh V., “On the rationality problem for conic bundles”, Duke Math. J., 54 (1987), 271–294 | DOI | MR | Zbl

[87] Iskovskikh V., “Towards the problem of rationality of conic bundles”, Lect. Notes Math., 1479, 1991, 50–56 | MR | Zbl

[88] Iskovskikh V., Prokhorov Yu., Fano varieties, Encyclopaedia Math. Sci., 47, Springer, Berlin, 1999 | MR | Zbl

[89] Iskovskikh V., Pukhlikov A., “Birational automorfisms of multi-dimentional algebraic varieties”, J. Math. Sci., 82:4 (1996), 3528–3613 | DOI | MR | Zbl

[90] J. de Jong A., Shepherd-Barron N., de Ven A. V., “On the Burkhardt quartic”, Math. Ann., 286 (1990), 309–328 | DOI | MR | Zbl

[91] Kawamata Yu., “A generalization of Kodaira–Ramanujam's vanishing theorem”, Math. Ann., 261 (1982), 43–46 | DOI | MR | Zbl

[92] Kawamata Yu., “Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces”, Ann. Math., 127 (1988), 93–163 | DOI | MR | Zbl

[93] Kawamata Yu., Matsuda K., Matsuki K., “Introduction to the minimal model problem”, Adv. Stud. Pure Math., 10 (1987), 283–360 | MR | Zbl

[94] Kollár J., “Nonrational hypersurfaces”, J. Amer. Math. Soc., 8 (1995), 241–249 | DOI | MR | Zbl

[95] Kollár J., Rational curves on algebraic varieties, Springer-Verlag, Berlin, 1996 | MR

[96] Kollár J., “Singularities of pairs. Algebraic geometry”, Proc. Summer Reserach Institute (Santa Cruz, CA, USA. July 9–29, 1995), 62, Amer. Math. Soc., Providence, RI, 1997, 221–287 | MR | Zbl

[97] Kollár J., “Low degree polynomial equations: arithmetic, geometry and topology”, Prog. Math., 168 (1998), 255–288 | MR | Zbl

[98] Kollár J., “Non-rational covers of $\mathbb{CP}^n\times\mathbb{CP}^m$”, L. M. S. Lecture Note Series, 281, 2000, 51–71 | MR | Zbl

[99] Kollár J. et al., “Flips and abundance for algebraic threefolds”, A summer seminar at the university of Utah (Salt Lake City, 1991), Astérisque, 211, 1992 | MR

[100] Marchisio M., “Unirational quartic hypersurfaces”, Boll. Unione Mat. Ital., 3 (2000), 301–314 | MR

[101] Matsusaka T., “Algebraic deformations of polarized varieties”, Nagoya Math. J., 31 (1968), 185–245 | MR | Zbl

[102] Mella M., “Birational geometry of quartic 3-folds. II: the importance of being $\mathbb Q$-factorial”, Math. Ann., 330:1 (2004), 107–126 | DOI | MR | Zbl

[103] Mori S., “Threefolds whose canonical bundles are not numerically effective”, Ann. Math., 115 (1982), 133–176 | DOI | MR

[104] Mori S., Mukai S., “Classification of Fano 3-folds with $B_2\geqslant 2$”, Manuscr. Mat., 36 (1981), 147–162 | DOI | MR | Zbl

[105] Mori S., Mukai S., “Erratum, Classification of Fano 3-folds with $B_2\geqslant 2$”, Manuscr. Mat., 110 (2003), 407 | DOI | MR

[106] Morrison D. R., “The birational geometry of surfaces with rational double points”, Math. Ann., 271 (1985), 415–438 | DOI | MR | Zbl

[107] Mukai S., “New developments of Fano varieties – on vector bundles and moduli problems”, Sugaku, 47:2 (1995), 125–144 | MR | Zbl

[108] Mukai S., Umemura H., “Minimal rational threefolds”, Lect. Notes Math., 1016, 1983, 490–518 | MR | Zbl

[109] Namikawa Y., “Smoothing Fano 3-folds”, J. Algebr. Geom., 6 (1997), 307–324 | MR | Zbl

[110] Prokhorov Yu., On the degree of Fano 3-folds with canonical Gorenstein singularities, arXiv: math.AG/0405347

[111] Pukhlikov A., “Birational automorphisms of double spaces with sigularities”, J. Math. Sci., 85 (1997), 2128–2141 | DOI | MR | Zbl

[112] Reid M., Projective morphism according to Kawamata, Preprint, University of Warwick, 1983 ; www.maths.warwick.ac.uk/~miles/3folds/Ka.ps | Zbl | Zbl

[113] Reid M., “Canonical 3-folds”, Journées de Géometrie Algébrique d'Angers, ed. A. Beauville, Sijthoff Noordhof, Alphen, 1980, 273–310 | MR

[114] Reid M., “Chapters on algebraic surfaces”, Complex algebraic geometry, Lecture notes from a summer program (Park City, Utah, 1993), ed. J. Kollár, 1997, 5–159 | MR | Zbl

[115] Roth L., Algebraic threefolds with special regard to problems of rationality, Springer-Verlag, Berlin, 1955 | MR

[116] Saint-Donat B., “Projective models of K-3 surfaces”, Amer. J. Math., 96 (1974), 602–639 | DOI | MR | Zbl

[117] Sano T., “Classification of non-Gorenstein $\mathbb Q$-Fano $d$-folds of Fano index greater than $d-2$”, Nagoya Math. J., 142 (1996), 133–143 | MR | Zbl

[118] Segre B., “Variazione continua ad omotopia in geometria algebrica”, Ann. Mat. Pura et Appl., 50 (1960), 149–186 | DOI | MR | Zbl

[119] Stevens J., “On canonical singularities as total spaces of deformations”, Abh. Math. Sem. Univ. Hamburg, 58 (1988), 275–283 | DOI | MR | Zbl

[120] Viehweg V., “Vanishing theorems”, J. Reine Angew. Math., 335 (1982), 1–8 | MR | Zbl

[121] Voisin C., Hodge theory and complex algebraic geometry, II, Cambridge Studies in Advanced Mathematics, 77, Cambridge Univ. Press, 2003 | MR | Zbl