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Algebraic Surfaces

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Algebraic Geometry II

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 35))

Abstract

The aim of this survey is to present a cohesive picture of the theory of algebraic surfaces, explain its problems, and describe its main methods. The proofs, when they are given, serve only to clarify the principal ideas employed in the field. For detailed proofs the reader is referred to the articles listed at the end of the survey.

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References

  • Ahlfors, L., Bers, L. [1961]: Riemann Surfaces and Quasiconformal Maps. Inostranaya Literatura: Moscow, 1961 (Russian)

    Google Scholar 

  • Algebraic Surfaces [1965]: Tr. Mat. Inst. Steklov. 75 (1965). English transi.: Proc. Steklov Inst. Math. 175 (1967), Zbl. 154, 210

    Google Scholar 

  • Algebraic Surfaces [1981]: C.I.M. 1977. Liguori: Napoli, 1981

    Google Scholar 

  • Aoki, N., Shioda, T. [1983]: Generators of the Neron-Severi group of a Fermat surface. In: Arithmetic and Geometry, I. Birkhäuser: Boston, 1983, Prog. Math. 35, 1–12, Zbl. 586.14028

    Google Scholar 

  • Artin, M. [1962]: Some numerical criteria for contractibility of curves on algebraic surfaces. Am. J. Math. 84 (1962) 485–496, Zbl. 105,144

    Article  MathSciNet  MATH  Google Scholar 

  • Artin, M., Mumford, D. [1972]: Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc., III. Ser. 25 (1972) 75–95, Zbl. 244.14017

    Article  MathSciNet  MATH  Google Scholar 

  • Artin, M., Winters, G. [1971]: Degenerate fibers and stable reduction of curves. Topology 11 (1971) 373–383, Zbl. 221.14018

    Article  MathSciNet  Google Scholar 

  • Barlow, R.N. [1982]: Some new surfaces with p g = 0. Thesis. Warwick, 1982. [See Duke Math. J. 51 (1984) 889–904], Zbl. 576.14038

    Google Scholar 

  • Barth, W., Peters, C., Van de Ven, A. [1984]: Compact complex surfaces. Springer-Verlag: Berlin Heidelberg New York, 1984, Zbl. 718.14023

    Book  MATH  Google Scholar 

  • Beauville, A. [1978]: Surfaces algébriques complexes. Asterisque 54 (1978) 1–172, Zbl. 394.14014

    MATH  Google Scholar 

  • Beauville, A. [1983]: Variétés Kählériennes dont la premiere class de Chern est nulle. J. Differ. Geometry 18 (1983) 755–782, Zbl. 537.53056

    MathSciNet  MATH  Google Scholar 

  • Bombieri, E. [1973]: Canonical models for surfaces of general type. Inst. Haut. Etud. Sci., Publ. Math. 42 (1973) 171–219, Zbl. 259.14005

    MathSciNet  Google Scholar 

  • Bombieri, E., Husemöller, D. [1975]: Classification and embeddings of surfaces. In: Algebraic Geometry (Arcata 1974). Am. Math. Soc., Proc. Symp. Pure Math. 29 (1975) 329–420, Zbl. 326.14009

    Article  Google Scholar 

  • Bombieri, E., Mumford, D. [1969, 1977, 1976]: Enriques’ classification of surfaces in characteristic p. I. In: Global analysis. Princeton Univ. Press: Princeton, 1969, 325–339. II. In: Complex analysis and algebraic geometry. Cambridge Univ. Press: Cambridge, 1977, 23–42. III. Invent. Math. 35 (1976) 197–232. Zbl. 188 532; Zbl. 348.14021; Zbl. 336.14010

    Google Scholar 

  • Bourbaki, N. [1968]: Eléments de mathématiques. Groupes et algèbres de Lie. Chapitres 4, 5, 6. Hermann: Paris, 1968, Zbl. 186,330

    Google Scholar 

  • Castelnuovo, G., Enriques, F. [1914]: Die algebraischen Flächen vom Gesichtspunkt der birationalen Transformationen aus. In: Enz. der mathematischen Wissenschaften, III, Heft 6, 1914, 674–768, Jbuch 45, 883

    Google Scholar 

  • Chen, Z. [1987]: On the geography of surfaces. Math. Ann. 277 (1987) 141–164, Zbl. 609.14023

    Article  MathSciNet  Google Scholar 

  • Clemens, C., Griffiths, Ph. [1972]: The intermediate Jacobian of the cubic threefold. Ann. Math., II. Ser. 95 (1972) 281–356, Zbl. 231.14004

    Article  MathSciNet  MATH  Google Scholar 

  • Danilov, V. I. [1988]: Algebraic Varieties and Schemes. Itogi Nauki Tekh. VINITI, Ser. Sovrem. Probi. Mat., Fundam. Napravleniya 23 (1988) 172–302. English transi.: Encycl. Math. Sci. 23. Springer-Verlag: Berlin Heidelberg New York, 1994, 167–297, Zbl. 787.14001

    Google Scholar 

  • Danilov, V. I. [1989]: Cohomologies of Algebraic Varieties. Itogi Nauki Teich. VINITI, Ser. Sovrem. Probi. Mat., Fundam. Napravleniya 35 (1989) 5–130. English transi.: Encycl. Math. Sci. 35. Springer-Verlag: Berlin Heidelberg New York, 1995 (in this volume)

    Google Scholar 

  • de Rham, G. [1955]: Variétés differentiables. Hermann: Paris, 1955 (2nd ed. 1960; Zbl. 89, 81)

    Google Scholar 

  • Dolgachev, I. V. [1966a]: Rational surfaces with a pencil of elliptic curves. Izv. Akad. Nauk SSSR, Ser. Mat. 30 (1966) 1073–1100 (Russian), Zbl. 187,187

    MathSciNet  MATH  Google Scholar 

  • Dolgachev, I. V. [1966b]: On Severi conjecture on simply connected algebraic surfaces. Dokl. Akad. Nauk SSSR 170 (1966) 249–252. English transi.: Sov. Math., Dokl. 7 (1966) 1169–1172, Zbl. 173, 229

    Google Scholar 

  • Enriques, F. [1949]: Le superficie algebriche. Zanichelli: Bologna, 1949, Zbl. 36, 371 Gieseker, D. [1977]: Global moduli for surfaces of general type. Invent. Math. 43 (1977) 233–282, Zbl. 389.14006

    Google Scholar 

  • Gizatulin, M. Kh. [1984]: Defining relations for the Cremona transformations of the plane. Izv. Akad. Nauk SSSR, Ser. Mat. 46 (1984) 909–970. English transi.: Math. USSR, Izv. 21 (1983) 211–268, Zbl. 509.14011

    Google Scholar 

  • Griffiths, Ph., Harris, J. [1978]: Principles of Algebraic Geometry. A Wiley-Interscience Publication: New York, 1978, Zbl. 408.14001

    MATH  Google Scholar 

  • Grothendieck, A. [1962]: Fondements de la Géométrie Algébrique. Secrétariat Math.: Paris, 1962, Zbl. 239.14002

    Google Scholar 

  • Hartshorne, R. [1977]: Algebraic Geometry. Springer-Verlag: Berlin Heidelberg New York, 1977

    Book  MATH  Google Scholar 

  • (3rd printing 1983 Springer), Zbl. 367.14001

    Google Scholar 

  • Hirzebruch, F. [1987]: Collected Papers. Vol. I. Springer-Verlag: Berlin Heidelberg New York, 1987, 345–360, Zbl. 627.01044

    MATH  Google Scholar 

  • Holzapfel, R.-P. [1980]: A class of minimal surfaces in the unknown region of surface geography. Math. Nachr. 98 (1980) 211–232, Zbl. 474.14022

    Article  MathSciNet  MATH  Google Scholar 

  • Horikawa, E. [1978]: On the periods of Enriques surfaces. I. Math. Ann. 234 (1978) 73–88; II. Math. Ann. 235 (1978) 217–246, Zbl. 371.14019; Zbl. 412.14015

    Google Scholar 

  • Igusa, J.-I. [1955]: On some problems in abstract algebraic geometry. Proc. Natl. Acad. Sci. USA 41 (1955) 964–967, Zbl. 67, 391

    Article  MathSciNet  MATH  Google Scholar 

  • Iskovskikh, V. A. [1985]: A proof of the theorem on relations in the two-dimensional Cremona group. Usp. Mat. Nauk 40 (1985) No. 5, 255–256. English transi.: Russ. Math. Surv. 40 (1985), No.5, 231–232, Zbl. 613.14012

    Google Scholar 

  • Kanev, V. [1987]: Spectral curves, simple Lie algebras and Prym-Tjurin varieties. [Preprint. Inst. Math. Bulgarian Acad. Sci., 1987] Proc. Symp. Pure Math. 49 (1989), Part 1, 627–645, Zbl. 711.14026

    Google Scholar 

  • Kawamata, Y. A. [1982]: A generalization of Kodaira-Ramanujam’s vanishing theorem. Math. Ann. 261 (1982) 43–46, Zbl. 488.14003

    Article  MathSciNet  MATH  Google Scholar 

  • Kawamata, Y. A., Matsuda, K., Matsuki, K. [1987]: Introduction to the minimal model problem. In: Proc. Symp. Alg. Geom. (Sendai 1985). Adv. Stud. Pure Math. 10, North-Holland: Tokyo, 1987, 283–360, Zbl. 672.14006

    Google Scholar 

  • Kodaira, K. [1960, 1963]: On compact analytic surfaces. I. Ann. Math., II. Ser. 71 (1960) 11–152; II. ibid. 77 (1960) 563–626; III. ibid. 78 (1963) 1–40, Zbl. 98, 130; Zbl. 118,158; Zbl. 171,196

    Google Scholar 

  • Kodaira, K. [1964–1969]: On the structure of compact complex analytic surfaces. I. Am. J. Math. 86 (1964) 751–798; II. Am. J. Math. 88 (1966) 682–721; III. Am. J. Math. 90 (1969) 55–83; IV. Am. J. Math. 90 (1969) 1048–1066. I: Zbl. 137, 175; II-IV: Zbl. 193, 377

    Google Scholar 

  • Kulikov, V. S. [1977]: Degenerations of K3 surfaces and Enriques surfaces. Izv. Akad. Nauk SSSR, Ser. Mat. 41 (1977) 1008–1042. English transl.: Math. USSR, Izv. 11 (1977) 957–989, Zbl. 367.14014

    Google Scholar 

  • Manin, Yu. I. [1972]: Cubic forms: Algebra, Geometry, Arithmetic. Nauka: Moscow, 1972. English. transl.: North-Holland: Amsterdam, 1974, 2nd ed. 1986, Zbl. 255.14002

    Google Scholar 

  • Milne, J. S. [1980]: Étale Cohomology. Princeton Math. Ser. 33, Princeton Univ. Press: Princeton, 1980, Zbl. 433.14012

    Google Scholar 

  • Milnor, J. [1963]: Morse theory. Princeton Univ. Press: Princeton, 1963, Zbl. 108, 104

    MATH  Google Scholar 

  • Miyaoka, Y. [1977]: On the Chern numbers of surfaces of general type. Invent. Math. 42 (1977) 225–237, Zbl. 374.14007

    Article  MathSciNet  MATH  Google Scholar 

  • Mori, S. [1982]: Threefolds whose canonical bundles are not numerically effective. Ann. Math., II. Ser. 115 (1982) 133–176, Zbl. 557.14021

    Article  Google Scholar 

  • Mori, S. [1987]: Classification of higher-dimensional varieties. In: Algebraic Geometry (Bowdoin 1985). Am. Math. Soc. Proc. Symp. Pure Math. 46, Part 1, 1987, 269–331, Zbl. 656.14022

    Google Scholar 

  • Mukai, S. [1988]: Finite groups of automorphisms of K 3 surfaces and the Mathieu group. Invent. Math. 94 (1988) 183–221, Zbl. 705.14045

    Article  MathSciNet  Google Scholar 

  • Mumford, D. [1966]: Lectures on Curves on an Algebraic Surface. Princeton Univ. Press: Princeton, 1966, Zbl. 187,427

    MATH  Google Scholar 

  • Mumford, D. [1970a]: Abelian Varieties. (Reprint, TIFR, Stud. Math. 5, Oxford 1985) Oxford Univ. Press: London, 1970, 1974, Zbl. 223.14022/Zbl. 326.14012

    Google Scholar 

  • Mumford, D. [1970b]: An algebraic surface with K ample, K 2 = 9, p = q = 0. Am. J. Math. 101 (1970) 233–244, Zbl. 433.14021

    Article  MathSciNet  Google Scholar 

  • Mumford, D. [1977]: Stability of projective varieties. Enseign. Math., II. Ser. 23 (1977) 39–110, Zbl. 363.14003

    MathSciNet  MATH  Google Scholar 

  • Mumford, D., Fogarty, J. [1982]: Geometric Invariant Theory, 2nd ed. Springer-Verlag: Berlin Heidelberg New York, 1982, Zbl. 504.14008 (3rd enl. ed. Springer 1993, Zbl. 797.14004)

    Book  MATH  Google Scholar 

  • Nikulin, V. V. [1981]: On quotient groups of automorphism groups of hyperbolic forms by subgroups generated by 2-reflections. Algebra-geometric applications. Itogi Nauki Tekh. VINITI, Ser. Sovrem. Probl. Mat. 18 (1981) 1–114, Zbl. 484.10021. English transl.: J. Soy. Math. 22 (1983) 1401–1475

    Google Scholar 

  • Nikulin, V. V. [1984]: The K 3 surfaces with finite automorphism group and rank 3 Picard group. Tr. Mat. Inst. Steklov 165 (1984) 119–142. English transl.: Proc. Steklov Inst. Math. 165 (1985) 131–155, Zbl. 577.10019

    Google Scholar 

  • Ogg, A. [1962]: Cohomology of Abelian varieties over function fields. Ann. Math., II. Ser. 76 (1962) 185–212, Zbl. 121,380

    Article  MathSciNet  MATH  Google Scholar 

  • Piatetski-Shapiro, I. I., Shafarevich, I. R. [1971]: A Torelli theorem for algebraic surfaces of type K 3. Izv. Akad. Nauk SSSR, Ser. Mat. 35 (1971) 530–572. English transl.: Math. USSR, Izv. 5 (1971) 547–588, Zbl. 219.14021

    Google Scholar 

  • Ramanan, S. [1978]: Vector bundles over algebraic curves. In: Proc. Int. Congr. Math., Helsinki 1978, Vol. II. 543–547 (1980), Zbl. 438.14014

    Google Scholar 

  • Ramanujam, C. P. [1978]: A tribute. Springer-Verlag: Berlin Heidelberg New York, 1978. Tata Inst. Fund. Res., Studies in Math. 8, Zbl. 401.00005 (entire collection)

    Google Scholar 

  • Reider, I. [1988]: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math., II. Ser. 127 (1988) 309–316, Zbl. 663.14010

    Article  MathSciNet  Google Scholar 

  • Saltman, D. J. [1984]: Noether’s problem over an algebraically closed field. Invent. Math. 77 (1984) 71–84, Zbl. 546.14014

    Article  MathSciNet  MATH  Google Scholar 

  • Seminar Palaiseau [1985]: Géométrie des surfaces K3: modules et periodes. Asterisque 126 (1985) 1–192, Zbl. 547.00019 (entire collection)

    Google Scholar 

  • Serre, J-P. [1970]: Cours d’arithmétique. Presses Univ. France: Paris, 1970. Zbl. 225.12002

    MATH  Google Scholar 

  • Shabat, G. B. [1977]: On complex structure of domains covering algebraic surfaces. Funkt. Anal. Appl. 11 (1977) 67–75. English transi.: Funct. Anal. Appl. 11 (1977) 135–142, Zbl. 355.32031

    Google Scholar 

  • Shafarevich, I. R. [1961]: Principal homogeneous spaces defined over function fields. Tr. Mat. Inst. Steklov. 64 (1961) 316–346, Zbl. 129,128. English transl.: Am. Math. Soc. Transl., II. Ser. 37 (1964) 85–114

    Google Scholar 

  • Shafarevich, I. R. [1988]: Basic Algebraic Geometry, Vol. I, II; 2nd suppl. ed. Nauka: Moscow, 1988. English transi.: Springer-Verlag: Berlin Heidelberg New York, 1994, Zbl. 675.14001 (Zbl. 797.14001, Zbl. 797.14002)

    Google Scholar 

  • Shioda, T. [1974]: An example of a unirational surface in characteristic p. Math. Ann. 211 (1974) 233–236, Zbl. 283.14009

    Article  MathSciNet  MATH  Google Scholar 

  • Shokurov, V. V. [1988]: Riemann surfaces and algebraic curves. Itogi Nauki Tekh. VINITI, Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 23 (1988) 5–171. English transi.: Encycl. Math. Sci. 23. Springer-Verlag: Berlin Heidelberg New York, 1994, 1–166, Zbl. 787.14022

    Google Scholar 

  • Siegel, C. L. [1949]: Analytic functions of several complex variables. Lecture Notes. Inst. Adv. Studies.: Princeton, 1949, Zbl. 36, 50

    Google Scholar 

  • Van de Ven, A. [1978]: Some recent results on surfaces of general type. In: Seminar Bourbaki, Exp. 500, 1976–1977. Lect. Notes Math. 677. Springer-Verlag: Berlin Heidelberg New York, 1978, 155–166, Zbl. 432.14023

    Google Scholar 

  • Van de Ven, A. [1987]: On the differentiable structure of certain algebraic surfaces. In: Seminar Bourbaki, Exp. 667, 1985–1986. Astérique 145–146 (1987) 299–312, Zbl. 624.57018

    Google Scholar 

  • Van der Geer, G. [1988]: Hilbert modular surfaces. Springer-Verlag: Berlin Heidelberg New York, 1988, Zbl. 634.14022

    Google Scholar 

  • Verra, E. [1983]: On the Enriques’ surfaces as a fourfold cover of P2. Math. Ann. 266 (1983) 241–250, Zbl. 519.14027

    Article  MathSciNet  MATH  Google Scholar 

  • Viehweg, E. [1982]: Vanishing theorems. J. Reine Angew. Math. 335 (1982) 1–8, Zbl. 485.32019

    MathSciNet  MATH  Google Scholar 

  • Vladimirov, V. S., Sergeev, A. G. [1985]: Complex analysis in the future tube. Itogi Nauki Tekh. VINITI, Ser. Sovrem. Probi. Mat., Fundam. Napravleniya 8 (1985) 191–266. English transl.: Encycl. Math. Sci. 8, 179–253. Springer-Verlag: Berlin Heidelberg New York, 1994, 179–254, Zbl. 787.32001

    Google Scholar 

  • Yau, S.-T. [1977]: On Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. USA 74 (1977) 1798–1799, Zbl. 355.32028

    Article  MATH  Google Scholar 

  • Zariski, O. [1958]: Introduction to the problem of minimal models in the theory of algebraic surfaces. Publ. Math. Soc. Japan 4 (1958) 1–89, Zbl. 93,339

    MathSciNet  Google Scholar 

  • Zariski, O. [1962]: The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Appendix by D. Mumford. Ann. Math., II. Ser. 76 (1962) 560–615, Zbl. 124,370

    Article  MathSciNet  MATH  Google Scholar 

  • Zariski, O. [1971]: Algebraic surfaces. 2nd ed. Springer-Verlag: Berlin Heidelberg New York, 1971, (1st ed. 1935, Zbl. 10,371), Zbl. 219.14020

    Google Scholar 

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Iskovskikh, V.A., Shafarevich, I.R. (1996). Algebraic Surfaces. In: Shafarevich, I.R. (eds) Algebraic Geometry II. Encyclopaedia of Mathematical Sciences, vol 35. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60925-1_2

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