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Inégalités numériques pour les surfaces de type général. Appendice: ”L’inégalité \(p_ g\geq 2q-4\) pour les surfaces de type général” par A. Beauville. (French) Zbl 0543.14026
The aim of this article is to improve Noether’s classical inequality: \(K^ 2\geq 2p_ g-4,\) for minimal complex algebraic surfaces of general type, by using the irregularity of the surface. We show the following inequalities:
\(K^ 2\geq 2p_ g+(q-4),\) with equality only for the product of two curves of genus 2; \(K^ 2\geq 2p_ g,\) if the surface is irregular. They are derived from the following inequalities: \(K^ 2\geq 3p_ g-2q+9,\) if the canonical system is composed with a pencil; \(K^ 2\geq 3p_ g+q-7\), if the canonical map is birational [F. Jongmans, Mém. Soc. Sci. Liège, IV. Ser.7, 367-468 (1947; Zbl 0036.377)]; \(K^ 2\geq 2p_ g+4(q-4)\), if the canonical map is of degree 2, with equality only for the product of two curves; one of them of genus 2, the other one of genus at least 2; \(K^ 2\geq 3p_ g-4,\) if the canonical map is of degree 3 and the surface is irregular.
Finally, an appendix written by A. Beauville is devoted to the proof of the inequality \(p_ g\geq 2q-4.\)
In a later addendum [Bull. Soc. Math. Fr. 111, 301-302 (1983)] a proof, due to Xiao Gang, of the following result is given: \(K^ 2\geq 2p_ g\), if the surface is irregular and the canonical system is composed with a pencil.

MSC:
14J25 Special surfaces
14N05 Projective techniques in algebraic geometry
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[1] BEAUVILLE (A.) . - Surfaces algébriques complexes , Astérisque, n^\circ 54, 1978 . MR 58 #5686 | Zbl 0394.14014 · Zbl 0394.14014
[2] BEAUVILLE (A.) . - L’application canonique pour les surfaces de type général , Invent. Math., vol. 55, 1979 , p. 121-140. MR 81m:14025 | Zbl 0403.14006 · Zbl 0403.14006 · doi:10.1007/BF01390086 · eudml:186129
[3] BOMBIERI (E.) . - Canonical models of surfaces of general type , Publ. Math. IHES, vol. 42, 1973 , p. 171-219. Numdam | MR 47 #6710 | Zbl 0259.14005 · Zbl 0259.14005 · doi:10.1007/BF02685880 · numdam:PMIHES_1973__42__171_0 · eudml:103921
[4] GRIFFITHS (P.) et HARRIS (J.) . - Algebraic Geometry , New York, John Wiley, 1978 . MR 80b:14001 | Zbl 0408.14001 · Zbl 0408.14001
[5] HORIKAWA (E.) . - On deformations of quintic surfaces , Invent. Math., vol. 31, 1975 , p. 43-85. Zbl 0317.14018 · Zbl 0317.14018 · doi:10.1007/BF01389865 · eudml:142355
[6] HORIKAWA (E.) . - Algebraic surfaces of general type with small c21 . Part I : Ann. of Math., vol. 104, 1976 , p. 357-387. MR 54 #12789 | Zbl 0339.14024 · Zbl 0339.14024 · doi:10.2307/1971050 · eudml:142430
[7] HORIKAWA (E.) . - Algebraic surfaces of general type with small c21 . Part II : Invent. Math., vol. 37, 1976 , p. 121-155. MR 57 #334 | Zbl 0339.14025 · Zbl 0339.14025 · doi:10.1007/BF01418966 · eudml:142430
[8] HORIKAWA (E.) . - Algebraic surfaces of general type with small c21 . Part III : Invent. Math., vol. 47, 1978 , p. 209-248. MR 80h:14012a | Zbl 0409.14005 · Zbl 0409.14005 · doi:10.1007/BF01579212 · eudml:142577
[9] HORIKAWA (E.) . - On algebraic surfaces with pencils of curves of genus 2 . Complex Analysis and Algebraic Geometry, a volume dedicated to K. KODAIRA, p. 79-90, Tokyo and Cambridge : Iwanami Shoten, Publishers and Cambridge University Press, 1977 . MR 56 #12015 | Zbl 0349.14021 · Zbl 0349.14021
[10] HARTSHORNE (R.) . - Algebraic Geometry , New York, Springer-Verlag, 1977 . MR 57 #3116 | Zbl 0367.14001 · Zbl 0367.14001
[11] JONGMANS (F.) . - Contribution à l’étude des variétés algébriques , Mém. Soc. R. Sc. Liège, vol. 7, 1947 , p. 367-468. MR 9,611c | Zbl 0036.37705 · Zbl 0036.37705
[12] NAGATA (M.) . - On rational surfaces I , Mem. coll. Sc. Univ. Kyoto, Ser. A, vol. 32, 1960 , p. 351-370. Article | MR 23 #A3739 | Zbl 0100.16703 · Zbl 0100.16703 · minidml.mathdoc.fr
[13] PERSSON (U.) . - On degenerations of algebraic surfaces , Mem. Amer. Math. Soc., vol. 189, 1977 . MR 57 #6030 | Zbl 0368.14008 · Zbl 0368.14008
[14] REID (M.) . - \pi 1 for surfaces with small c21 , Springer Lecture Notes n^\circ 732, Proceedings Copenhagen, 1978 . MR 81j:14018 | Zbl 0423.14021 · Zbl 0423.14021
[15] BEAUVILLE (A.) . - Surfaces algébriques complexes , Astérisque, n^\circ 54, 1978 . MR 58 #5686 | Zbl 0394.14014 · Zbl 0394.14014
[16] COMESSATTI (A.) . - Intorno alle superficie algebriche irregolari con pg\geq 2(pa+2) e ad un problema analitico ad esse collegato , Rend. Circ. Mat. di Palermo, vol. 46, 1922 , p. 1-48. JFM 48.0748.03 · JFM 48.0748.03
[17] CHAFAREVITCH (I.) et al. - Algebraic surfaces , Proc. of the Steklov Institute, vol. 75, 1965 .
[18] JONGMANS (F.) . - Sur l’étude des surfaces algébriques caractérisées par la condition pg\geq 2(pa+2) , Bull. Acad. Roy. de Belgique, s.5, 36, 1950 , p. 485-494. MR 12,739b | Zbl 0041.48101 · Zbl 0041.48101
[19] LANG (S.) . - Abelian varieties , Interscience Wiley, New York, 1959 . MR 21 #4959 | Zbl 0098.13201 · Zbl 0098.13201
[20] NOLLET (L.) . - Sur l’étude des surfaces algébriques caractérisées par la condition pg\geq 2(pa+2) , Bull. Acad. Roy. de Belgique, s.5, 36, 1950 , p. 897-905. MR 13,679f | Zbl 0040.23203 · Zbl 0040.23203
[21] ROSENBLATT (A.) . - Sur les surfaces irrégulières dont les genres satisfont à l’inégalité pg\geq 2(pa+2) , Rend. Circ. Mat. di Palermo, vol. 35, 1913 , p. 237-244. JFM 44.0715.02 · JFM 44.0715.02
[22] SZPIRO (L.) . - Sur le théorème de rigidité de Parsin et Arakelov . Journées de Géométrie algébrique de Rennes, Astérisque, n^\circ 64, 1979 , p. 169-202. MR 81f:14004 | Zbl 0425.14005 · Zbl 0425.14005
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