@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}, publisher = {Kluwer Academic Publishers}, volume = {85}, number = {1}, year = {1993}, mrnumber = {1199203}, zbl = {0780.14022}, language = {en}, url = {http://www.numdam.org/item/CM_1993__85_1_37_0/} }

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

[AltK1] Foundations of the theory of Fano schemes, Comp. Math. 34(1977), 3-47. | Numdam | MR | Zbl

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

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

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

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

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

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

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

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

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

,[ClCoMo] Higher dimensional complex geometry, Astérisque 166 (1988). | MR | Zbl

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

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

,[CoMuWe] On the family of conics lying on a quartic threefold, Rend. Sem. Mat. Univers. Polit. Torino 38 (1980), 151-181. | MR | Zbl

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

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

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

,[FraMaTschi] Rational points of bounded height on Fano varieties, Inv. Math. 95 (1989), 421-435. | MR | Zbl

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

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

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

,[Is2] 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). | MR | Zbl

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

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

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

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

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

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

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

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

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

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

and ,[MoMu2] 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

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

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

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

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

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

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

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

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

,[Se2] Comptes Rendus, 1990.

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

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

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

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

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

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

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

,[Tyu4] 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). | MR | Zbl

,[VoK1] 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

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

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

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

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

,