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Introduction to the Minimal Model Problem

Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.

Classical Algebraic Geometry: A Modern View

Introduction Chapter I. Intersection Numbers ? 1. The polynomial theorem of Snapper ? 2. The definition and some properties of intersection numbers ? 3. Degrees and Hilbert polynomials ? 4. Numerically effective and numerically trivial invertible sheaves Chapter II. Some Vanishing Theorems and Related Results ? 1. M-families ?2. Some vanishing theorems and related results Chapter III. Ampleness and Pseudoampleness ? 1. The numerical characterization of ampleness ? 2. Pseudoampleness Chapter IV. The Ample Cone ? 1. The vector space A'(V) ? 2. The ample cone ? 3. Polarization ? 4. Relativization ? 5. The characteristic cone of a proper morphism Bibliography

Toward a numerical theory of ampleness

[Ac] R. ACCOLA, Some loci of Teichmüller space for genus five defined by vanishing theta nulls , in "Contribution to analysis" (a collection of papers dedicated to L. Bers), Academic Press, New York. 1974, pp. 11-18. | MR 374411 | Zbl 0313.32029 [ACGH] E. ARBARELLO, M. CORNALBA, P. GRIFFITHS, J. HARRIS, Geometry of algebraic curves, vol. I, Springer -Verlag. 1984. | Zbl 0559.14017 [AS] Algebraic surfaces (ed. by I.Shafarevich), Proc.SteKlov.Math.Inst., v. 75. 1964. [Engl.transl: AMS, Providence . R.I. 1967]. | MR 190143 [Ba] H. BAKER, Principles of geometry, vol. IV. Frederick Unger Publ. New York. 1925 (republished by Cambridge Univ.Press. 1963). | MR 178393 [Be] A. BEAUVILLE, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. Ecole Norm. Sup. 10 (1977), 309-331. | Article | Numdam | MR 472843 | Zbl 0368.14018 [Bo] N. BOURBAKI, Groupes et algèbres de Lie. Chapters IV-VI. Herman. Paris. 1966 [Russ. transl.: Mir. Moscow. 1972]. | Zbl 0483.22001 [Bog] F. BOGOMOLOV, Stable rationality of factor spaces for simply connected groups, Mat. Sbornik 130 (1986), 3-17 [Engl. Transl.: Mathematics of USSR-Sbornik, 58 (1987), 1-17). | MR 847340 | Zbl 0615.14031 [Br ] G. BREDON. Introduction to compact transformation groups. Academic Press. 1972 [Russ. Transl.: Nauka. Moscow. 1980]. | MR 617738 | Zbl 0484.57001 [Ca] A. CAYLEY, A memoir on quartic surfaces . Proc. London Math.Soc, 3 (1869/71), 19-69, | JFM 03.0390.01 | MR 1577187 A. CAYLEY, A memoir on quartic surfacesCol. works, VII, Cambridge Univ. Press. 1894, 133-181. [Co 1] A. COBLE, Algebraic geometry and theta functions, Amer. Math. Soc. Coll. Publ. 10, Providence, R.I., 1929 (3d ed., 1969). | JFM 55.0808.02 | MR 733252 [Co 2] A. COBLE, Point sets and allied Cremona groups, I , Trans. Amer. Math. Soc. 16 (1915), 155-198. | JFM 45.0234.01 | MR 1501008 [Co 3] A. COBLE, Point sets and allied Cremona groups, II , Trans. Amer. Math. Soc. 17 (1916), 345-388. | JFM 45.0234.01 | MR 1501047 [Co 4] A. COBLE, Point sets and allied Cremona groups. III , Trans. Amer. Math. Soc. 18 (1917), 331-372. | JFM 46.0890.01 | MR 1501073 [Co 5] A. COBLE, Associated sets of points. Trans. Amer.Math.Soc. 24 (1922), 1-20. | Article | JFM 49.0490.01 | MR 1501210 [Co 6] A. COBLE, The ten nodes of the rational sextic and of the Cayley symmetroid . Amer. J. Math. 41 (1919), 243-265. | Article | JFM 47.0596.01 | MR 1506391 [Co 7] A. COBLE, Theta modular groups determined by point sets. Amer. J. Math., 40 (1918), 317-340. | Article | JFM 46.0612.04 | MR 1507902 [Co 8] A. COBLE, Configurations defined by theta functions, Duke Math. J., 5 (1939), 479-488. | Article | MR 160 | Zbl 0022.16304 [Coo] J. COOLIDGE, A treatise on algebraic plane curves. Dover Publ. 1959. | MR 120551 | Zbl 0085.36403 Bibliographie Rechercher des articles, des livres... Tout + 

Point sets in projective spaces and theta functions

This article contains a classification of special Fano varieties; we give a description of the projective models of Fano 3-folds of index r ≥ 2 and of hyperelliptic Fano 3-folds.Bibliography: 31 titles.

Fano 3-FOLDS. I

In §1 of the present paper, we introduce the notion of a virtual linear system on a non-singular projective surface and we clarify the theories of infinitely near points, of divisors and of linear system with preassigned base conditions. We introduce in §2 the notions of a numerical types and of non-special points with respect to Cremona transformations. They play important roles in §3 in order to prove characterizations and existence theorems of exceptional curves of the first kind and of Cremona transformations. In §4, we introduce the notion of an abnormal curve, and in §5 we give some remarks on superabundance of a complete virtual linear system on a projective plane S. We add some remarks in §6 on the case where the number of base points is at most 9. The recent paper “On rational surfaces, I” in the last volume of our memoirs is quoted as Part I in the present paper. The notations and terminology in Part I are preserved in this paper, except for that the symbol { } for the total transform of a divisor is changed to ( ) ; see §1. We recall here that an $S$ denotes always a projective plane. A curve will mean a positive divisor on a surface. A divisor $c$ on a surface $F$ is identified with a divisor $c'$ on a surface $F'$ if $c=\sum m_{i}c_{i}$ and $c'=\sum m_{i}c'_{i}$ and if $c_{i}$ and $c'_{i}$ are irreducible and are identical with each other as point sets (identification of points is made by natural birational transformations).

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Masayoshi Nagata

Papers

On rational surfaces, II