A Remark on Algebraic Surfaces with Polyhedral Mori Cone

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 X ∈ FPMC is finite and generates NE(X). Let δE(X) be the maximum of (−C ) 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 X ∈ FPMC 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 ≤ N(ρ, δE) and h′ ≤ N (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 X ∈ FPMC as Algebraic Geometry analog of the Theory of arithmetic reflection groups in hyperbolic spaces.