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A remark on algebraic surfaces with polyhedral Mori cone

  • Viacheslav V. Nikulin (a1)
Abstract

We denote by FPMC the class of all non-singular projective algebraic surfaces X over ℂ with finite polyhedral Mori cone NE(X) ⊂ NS(X) ⊗ ℝ. If ρ(X) = rk NS(X) ≥ 3, then the set Exc(X) of all exceptional curves on XFPMC is finite and generates NE(X). Let δE(X) be the maximum of (-C 2) and pE(X) the maximum of pa(C) respectively for all C ∈ Exc(X). For fixed ρ ≥ 3, δE and pE we denote by FPMCρ,δE,pE the class of all algebraic surfaces XFPMC such that ρ(X) = ρ, δE(X) = δE and pE(X) = pE . We prove that the class FPMCρ,δE,pE is bounded in the following sense: for any X ∈ FPMCρ,δE,pE there exist an ample effective divisor h and a very ample divisor h′ such that h 2N(ρ, δE) and h2N′(ρ, δE, pE) where the constants N(ρ, δE) and N′(ρ, δE, pE) depend only on ρ, δE and ρ, δE, pE respectively.

One can consider Theory of surfaces XFPMC as Algebraic Geometry analog of the Theory of arithmetic reflection groups in hyperbolic spaces.

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References
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[AN1] Alexeev, V.A. and Nikulin, V.V., The classification of Del Pezzo surfaces with log terminal singularities of the index ≤ 2, involutions of K3 surfaces and reflection groups in Lobachevsky spaces (Russian), Doklady po matematike i prilogeniyam, MIAN, 2 (1988), no. 2, 51150.
[AN2] V.Alexeev, A. and Nikulin, V.V., The classification of Del Pezzo surfaces with log terminal singularities of the index ≤ 2 and involutions of K3 surfaces, Dokl. AN SSSR, 306 (1989), no. 3, 525528.
[B1] Borcherds, R., Automorphic forms on Os+2,2 and infinite products, Invent. Math., 120 (1995), 161213.
[B2] Borcherds, R., The moduli space of Enriques surfaces and the fake monster Lie super- algebra, Topology, 35 (1996), no. 3, 699710.
[CCL] Cardoso, G.L., Curio, G. and Löst, D., Perturbative coupling and modular forms in N = 2 string models with a Wilson line, Nucl. Phys., B491 (1997), 147183; hep-th/9608154.
[CD] F.Cossec, R. and Dolgachev, I.V., Enriques surfaces I, Birkhäuser, Progress in Mathematics, Vol. 76, 1989, pp. 397.
[D] Dolgachev, I.V., On rational surfaces with elliptic pencil (Russian), Izv. AN SSSR, Ser. matem., 30 (1966), 10731100.
[E] Esselmann, F., Über die maximale Dimension von Lorentz-Gittern mit coendlicher Spiegelungsgruppe, J. Number Theory, 61 (1996), 103144.
[GN] Gritsenko, V.A. and Nikulin, V.V., Siegel automorphic form correction of some Lorentzian Kac-Moody Lie algebras, Amer. J. Math., 119 (1997), no. 1, 181224; alg-geom/9504006.
[GN2] Gritsenko, V.A. and Nikulin, V.V., Siegel automorphic form correction of a Lorentzian Kac-Moody algebra, C. R. Acad. Sci. Paris Sér. A-B, 321 (1995), 11511156.
[GN3] Gritsenko, V.A. and Nikulin, V.V., K3 surfaces, Lorentzian Kac-Moody algebras and mirror symmetry, Math. Res. Lett., 3 (1996), no. 2, 211229; alg-geom/9510008.
[GN4] Gritsenko, V.A. and Nikulin, V.V., The Igusa modular forms and “the simplest” Lorentzian Kac-Moody algebras, Matem. Sbornik, 187 (1996), no. 11, 2766.
[GN5] Gritsenko, V.A. and Nikulin, V.V., Automorphic forms and Lorentzian Kac-Moody algebras. Part I, Intern. J. Math., 9 (1998), no. 2, 153199; alg-geom/9610022.
[GN6] Gritsenko, V.A. and Nikulin, V.V., Automorphic forms and Lorentzian Kac-Moody algebras. Part II, Intern. J. Math., 9 (1998), no. 2, 201275; alg-geom/9611028.
[GN7] Gritsenko, V.A. and Nikulin, V.V., The arithmetic mirror symmetry and Calabi-Yau manifolds, Commun. Math., to appear; alg-geom/9612002.
[Hal] Halphen, , Sur les courbes planes du sixiéme degré à neuf points doubles, Bull. Soc. math. France, 10 (1881), 162172.
[Har] Hartshorne, R., Algebraic Geometry, Springer, 1977, pp. 496.
[HM] Harvey, J. and Moore, G., Algebras, BPS-states, and strings, Nucl. Physics., B463 (1996), 315368; hep-th/9510182.
[Ka] Kawai, T., String duality and modular forms, Phys. Lett., B397 (1997), 5162; hep-th/9607078.
[Kaw] Kawamata, Y., A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann., 261 (1982), 4346.
[Ko] Kondo, S., Enriques surfaces with finite automorphism groups, Japanese J. Math., 12 (1986), no. 2, 191282.
[L] Lazarsfeld, R., Lectures on linear systems, Complex Algebraic Geometry (Kollar, J., eds.), Amer. Math. Soc. (IAS/Park City, Math. Series, Vol. 3) (1997).
[Ma] Manin, Yu.I., Cubic forms: Algebra, Geometry, Arithmetic, Nauka, 1972.
[Mo] Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. Math., 116 (1982), no. 1, 133176.
[Moo] Moore, G., String duality, automorphic forms, and generalized Kac-Moody algebras, Preprint (1997; hep-th/9710198).
[Na] Nagata, M., On rational surfaces I, II, Mem. of College of Sci. Univ. of Kyoto, Ser. A, 32 (1960), no, 3 351370.
[N1] Nikulin, V.V., Integral symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979), 111177.
[N2] Nikulin, V.V., On factor groups of the automorphism groups of hyperbolic forms modulo subgroups generated by 2-reflections, Dokl. Akad. Nauk SSSR, 248 (1979), 13071309.
[N3] Nikulin, V.V., On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2-reflections, Algebraic-geometric applications, Current Problems in Math. Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 18 (1981), 3114.
[N4] Nikulin, V.V., On arithmetic groups generated by reflections in Lobachevsky spaces, Math. USSR Izv., 16 (1981), 637669.
[N5] Nikulin, V.V., On the classification of arithmetic groups generated by reflections in Lobachevsky spaces, Math. USSR Izv., 18 (1982), 45 (1981), no. 1, 113142.
[N6] Nikulin, V.V., Surfaces of type K3 with finite automorphism group and Picard group of rank three, Trudy Inst. Steklov, 165 (1984), 113142.
[N7] Nikulin, V.V., On a description of the automorphism groups of Enriques surfaces, Dokl. AN SSSR, 277 (1984), 13241327.
[N8] Nikulin, V.V., Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, Proc. Int. Congr. Math. Berkeley 1986, 1, pp. 654669.
[N9] Nikulin, V.V., Basis of the diagram method for generalized reflection groups in Lobachevsky spaces and algebraic surfaces with nef anticanonical class, Intern. J. of Math., 7 (1996), no. 1, 71108.
[N10] Nikulin, V.V., A lecture on Kac-Moody Lie algebras of the arithmetic type, Preprint Queen’s University, Canada, #1994-16, (1994).
[N11] Nikulin, V.V., Reflection groups in Lobachevsky spaces and the denominator identity for Lorentzian Kac-Moody algebras, Izv. Akad. Nauk of Russia. Ser. Mat., 60 (1996), no. 2, 73106.
[N12] Nikulin, V.V., The remark on discriminants of K3 surfaces moduli as sets of zeros of automorphic forms, J. of Mathematical Sciences,, 81 (1996), no. 3, Plenum Publishing; alg-geom/9512018, 27382743.
[N13] Nikulin, V.V., K3 surfaces with interesting groups of automorphisms, J. of Math. Sciences, to appear; alg-geom/9701011.
[N14] Nikulin, V.V., On the classification of hyperbolic root systems of the rank three., Preprint Max-Planck-Institut für Mathematik,, MPI-99 (1999).
[O] Ogg, A., Cohomology of abelian varieties over function fields, Ann. Math., 76 (1962), 185212.
[P-ŠŠ] Pjatetckiĭ-Šapiro, I.I. and Šafarevich, I.R., A Torelli theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 530572.
[R] Reider, I., Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. Math., 127 (1988), 309316.
[Sh1] Šafarevič, I.R. (ed.), Algebraic surfaces, Proc. Steklov Math. Instit., 1965.
[Sh2] Šafarevič, I.R. (ed.) Basic Algebraic Geometry, Nauka, 1988.
[Sh3] Šafarevič, I.R. (ed.) Principal homogeneous spaces over function fields, Proc. Steklov Inst. Math., 64 (1961), 316346.
[Vie] Viehweg, E., Vanishing theorems, J. reine und angew. Math., 335 (1982), 18.
[V1] Vinberg, E.B., The absence of crystallographic reflection groups in Lobachevsky spaces of large dimension, Trudy Moscow. Mat. Obshch., 47 (1984), 67102.
[V2] Vinberg, E.B., Hyperbolic reflection groups, Uspekhi Mat. Nauk, 40 (1985), 2966.
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Nagoya Mathematical Journal
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