Notes on the arithmetic of Fano threefolds

Manin, Yu I.

Manin, Yu I.

Compositio Mathematica, Tome 85 (1993) no. 1, pp. 37-55

MR Zbl | 5 citations dans Numdam

@article{CM_1993__85_1_37_0,
     author = {Manin, Yu I.},
     title = {Notes on the arithmetic of {Fano} threefolds},
     journal = {Compositio Mathematica},
     pages = {37--55},
     year = {1993},
     publisher = {Kluwer Academic Publishers},
     volume = {85},
     number = {1},
     mrnumber = {1199203},
     zbl = {0780.14022},
     language = {en},
     url = {https://www.numdam.org/item/CM_1993__85_1_37_0/}
}

TY  - JOUR
AU  - Manin, Yu I.
TI  - Notes on the arithmetic of Fano threefolds
JO  - Compositio Mathematica
PY  - 1993
SP  - 37
EP  - 55
VL  - 85
IS  - 1
PB  - Kluwer Academic Publishers
UR  - https://www.numdam.org/item/CM_1993__85_1_37_0/
LA  - en
ID  - CM_1993__85_1_37_0
ER  -

%0 Journal Article
%A Manin, Yu I.
%T Notes on the arithmetic of Fano threefolds
%J Compositio Mathematica
%D 1993
%P 37-55
%V 85
%N 1
%I Kluwer Academic Publishers
%U https://www.numdam.org/item/CM_1993__85_1_37_0/
%G en
%F CM_1993__85_1_37_0

Manin, Yu I. Notes on the arithmetic of Fano threefolds. Compositio Mathematica, Tome 85 (1993) no. 1, pp. 37-55. https://www.numdam.org/item/CM_1993__85_1_37_0/

Bibliographie
Cité par

[AltK1] A.B. Altman and S.I. Kleiman, Foundations of the theory of Fano schemes, Comp. Math. 34(1977), 3-47. | Zbl | MR | Numdam

[Ba] V.V. Batyrev, Toroidal Fano threefolds, Math. USSR Izv. 19 (1982), 13-25. | Zbl

[BaMa] V.V. Batyrev And Yu. I. Manin, Sur le nombre des points rationnels de hauteur borné des variétés algébriques, Math. Annalen Bd. 286 (1990), 27-43. | Zbl | MR

[BarVe] W. Barth and A. van de Ven, Fano varieties of lines on hypersurfaces, Arch. Math. 31 (1978), 96-104. | Zbl | MR

[Beau] A. Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. ENS 10 (1977), 304-392. | Zbl | MR | Numdam

[BlMu] S. Bloch and J.P. Murre, On the Chow group of certain types of Fano threefolds, Comp. Math. 39 (1979), 47-105. | Zbl | MR | Numdam

[BoSp] A. Borel and T.A. Springer, Rationality properties of linear algebraic groups, in: Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math., vol. 9, 26-32, Providence RI, 1980. | Zbl | MR

[BoVe] L.P. Botta and A. Verra, The non-rationality of the generic Enriques threefold, Comp. Math. 48 (1983), 167-184. | Zbl | MR | Numdam

[CeVe] G. Ceresa and A. Verra, The Abel-Jacobi isomorphism for the sextic double solid, Pacific J. of Math. 124 (1986), 85-105. | Zbl | MR

[Cl] C.H. Clemens, Double solids, Adv. Math. 47 (1983), 107-230. | Zbl | MR

[ClCoMo] H. Clemens, J. Kollár and S. Mori, Higher dimensional complex geometry, Astérisque 166 (1988). | Zbl | MR | Numdam

[ClGr] H. Clemens And Ph. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1982), 281-356. | Zbl | MR

[Co] A. Collino, Lines on quartic threefolds, J. Lond. Math. Soc. ser. 2, 19 (1979), 257-267. | Zbl | MR

[CoMuWe] A. Collino, J.P. Murre and G. Welters, On the family of conics lying on a quartic threefold, Rend. Sem. Mat. Univers. Polit. Torino 38 (1980), 151-181. | Zbl | MR

[Dem1] M. Demazure, A very simple proof of Bott's theorem, Inv. Math. 33 (1976), 271-272. | Zbl | MR

[Dem2] M. Demazure, Automorphismes et déformations des variétés de Borel, Inv. Math. 39 (1977), 179-186. | Zbl | MR

[Do] R. Donagi, Group law on the intersection of two quadrics, Ann. Scuola Norm. Sup. di Pisa VII (1980), 217-239. | Zbl | MR | Numdam

[FraMaTschi] J. Franke, Yu. I. Manin and Yu. Tschinkel, Rational points of bounded height on Fano varieties, Inv. Math. 95 (1989), 421-435. | Zbl | MR

[Fu] T. Fujita, On the structure of polarized manifolds with total deficiency one, J. Math. Soc. Japan. I, 32 (1980); II, 33 (1981), 415-434; III, 36 (1984), 75-89. | Zbl

[Gu1] N.P. Gushel, On Fano varieties of genus 6, Izv. AN SSSR, ser. mat. 46 (1982), 1159-1174 (in Russian). | Zbl | MR

[Gu2] N.P. Gushel, On Fano varieties of genus 8, Uspekhi 38 (1983), 163-164 (in Russian). | Zbl | MR

[Is1] V.A. Iskovskih, Fano threefolds, I, II, Izv. AN SSSR 41 (1977), 516-562; 42 (1978), 504-549 (in Russian). | Zbl | MR

[Is2] V.A. Iskovskih, Anticanonical models of three-dimensional algebraic varieties, VINITI, Sovr. Probl. Mat. 12 (1979), 159-236 (in Russian); J. Sov. Math. 13, no. 6 (1980), 745-814 (in translation). | Zbl | MR

[Is3] V.A. Iskovskih, Birational automorphisms of three-dimensional algebraic varieties, ibid. 159-236 (in Russian); ibid. 815-868 (in translation). | Zbl | MR

[Is4] V.A. Iskovskih, Lectures on algebraic threefolds, Fano threefolds, Moscow University, 1988 (in Russian).

[Le] M. Letizia, The Abel-Jacobi mapping for the quartic threefold, Inv. Math. 75 (1984), 477-492. | Zbl | MR

[Ma] Yu. I. Manin, Cubic Forms, 2nd ed., North-Holland, Amsterdam, 1986. | Zbl | MR

[Mar] D.G. Markushevich, Numerical invariants of line families on certain Fano varieties, Mat. Sbornik 116 (1981), 265-288 (in Russian). | Zbl | MR

[Mar-Des] M. Martin-Deschamps, Propriétés de descente des variétés à fibre cotangent ample, Ann. Inst. Fourier 33 (1984), 39-64. | Zbl | MR | Numdam

[MaTschi] Yu. I. Manin and Yu. Tschinkel, Points of bounded height on del Pezzo surfaces, 1991 (preprint). | Zbl | Numdam

[Me] J.Y. Merindol, Théorème de Torelli affine pour les intersections de deux quadriques, Inv. Math. 80 (1985), 375-416. | Zbl | MR

[Mu] J.P. Murre, Classification of Fano threefolds according to Fano and Iskovskih, in: Algebraic threefolds, Proceedings, Varenna 1981 Lect. N. Math. 947, Springer-Verlag (1982), 35-92. | Zbl | MR

[MoMu1] Sh. Mori and Sh. Mukai, Classification of Fano 3-folds with B2 ≽ 2, Manuscripta Math. 36, no. 2 (1981), 147-162. | Zbl

[MoMu2] Sh. Mori and Sh. Mukai, On Fano threefolds with B2 ≽ 2, in: Advanced Studies in Pure Math. 1 (1983), Algebraic Varieties and Analytic Varieties, p. 101-129, Kinokuniya, Amsterdam-Tokyo, 1985. | Zbl

[MoMu3] Sh. Mori and Sh. Mukai, Classification of Fano 3-folds with B2 ≽ 2, in: Algebraic and Topological Theories, Kinokuniya, Amsterdam-Tokyo, 1985, 496-548. | Zbl

[Na] A.M. Nadel, The boundedness of degree of Fano varieties with Picard number one, 1990 (preprint MIT). | Zbl | MR

[New] P.E. Newstead, Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205-215. | Zbl | MR

[Pu1] A.V. Pukhlikov, Birational automorphisms of a three-dimensional quartic with the simplest singularity, Mat. Sbornik 135 (1988), 472-496 (in Russian). | Zbl | MR

[Pu2] A.V. Pukhlikov, Birational automorphisms of a double space and double quadric, Izv. AN SSSR, ser. mat. 52 (1988), 229-239 (in Russian). | Zbl | MR

[Re1] M. Reid, The complete intersection of two or more quadrics, Ph.D. These, Cambridge Univ., 1972.

[Re2] M. Reid, Lines on Fano threefolds according to Shokurov, Inst. Mittag Loeffler Rep. 11 (1980), 1-41.

[Se1] J.-P. Serre, Lectures on the Mordell-Weil theorem, Vieweg, Braunschweig-Wiesbaden, 1989. | Zbl | MR

[Se2] J.-P. Serre, Comptes Rendus, 1990.

[Sh1] V.V. Shokurov, Smoothness of the generic anticanonical divisor on Fano varieties, Izv. AN SSSR, ser. mat. 43 (1979), 430-441 (in Russian). | Zbl | MR

[Sh2] V.V. Shokurov, Existence of lines on Fano varieties, Izv. AN SSSR, ser. mat. 43 (1979), 922-964 (in Russian). | Zbl | MR

[Te] B. Tennison, On the quartic threefold, Proc. Lond. Math. Soc. 29 (1974), 714-734. | Zbl | MR

[Ti] A.S. Tikhomirov, The intermediate Jacobian of a double solid ramified at a quartic, Izv AN SSSR, ser. mat. 44 (1980), 1329-1377 (in Russian). | Zbl | MR

[Tyu1] A.N. Tyurin, Five lectures on threefolds, Uspekhi 27 (1972), 3-50 (in Russian). | MR

[Tyu2] A.N. Tyurin, On the intersections of quadrics, Uspekhi 30 (1975), 51-99 (in Russian). | Zbl | MR

[Tyu3] A.N. Tyurin, Intermediate Jacobians in the theory of threefolds, in: VINITI, Sovr. Prob. Mat. 12 (1979), 5-57 (in Russian). | Zbl

[Tyu4] A.N. Tyurin, Geometry of the Fano surface of a smooth cubic F c P4 and Torelli theorems for Fano surfaces and cubics, Izv. AN SSSR, ser. mat. 35 (1971), 498-529 (in Russian). | Zbl | MR

[VoK1] V.E. Voskresenskii and A.A. Klyachko, Toroidal Fano varieties and root systems, Izv. AN SSSR, ser. mat. 48 (1984) (in Russian); Math. USSR Izvestiya 24 (1985), 221-243 (in translation). | Zbl

[WaWa] K. Watanabe and M. Watanabe, The classification of Fano 3-folds with torus embeddings, Tokyo J. Math. 5 (1982), 37-48. | Zbl | MR

[We1] G. Welters, Abel-Jacobi isogenies for certain types of Fano threefolds, Math. Centrum Amsterdam, 1981. | Zbl | MR

[We2] G. Welters, The Fano surface of lines on a double P3 with 4th order discriminant locus, Part 1, 1979 (Utrecht Univ. preprint). | MR

[Wi] P.M.H. Wilson, Fano fourfolds of index greater than one, Journ. reine angew. Math. (Crelle) 379 (1987), 172-181. | Zbl | MR