A remark on algebraic surfaces with polyhedral Mori cone

Viacheslav V. Nikulin
2000 Nagoya mathematical journal  
We denote byFPMCthe class of all non-singular projective algebraic surfacesXover ℂ with finite polyhedral Mori cone NE(X) ⊂ NS(X) ⊗ ℝ. Ifρ(X) = rk NS(X) ≥ 3, then the set Exc(X) of all exceptional curves onX∈FPMCis finite and generates NE(X). LetδE(X) be the maximum of (-C2) andpE(X) the maximum ofpa(C) respectively for allC∈ Exc(X). For fixedρ≥ 3,δEandpEwe denote byFPMCρ,δE,pEthe class of all algebraic surfacesX∈FPMCsuch thatρ(X) =ρ, δE(X) =δEandpE(X) =pE. We prove that the classFPMCρ,δE,pEis
more » ... classFPMCρ,δE,pEis bounded in the following sense: for anyX ∈ FPMCρ,δE,pEthere exist an ample effective divisorhand a very ample divisorh′ such thath2≤N(ρ, δE) andh′2≤N′(ρ, δE, pE) where the constantsN(ρ, δE) andN′(ρ, δE, pE) depend only onρ, δEandρ, δE, pErespectively.One can consider Theory of surfacesX∈FPMCas Algebraic Geometry analog of the Theory of arithmetic reflection groups in hyperbolic spaces.
doi:10.1017/s0027763000007194 fatcat:rkqh6rk5yjbpjg4amrfbzery3q