A remark on algebraic surfaces with polyhedral Mori cone

Abstract

We denote by FPMC the class of all non-singular projective algebraic surfaces X over C with a finite polyhedral Mori cone NE(X)\subset NS(X)\otimes R. If rho(X)=rk NS(X)\ge 3, then the set Exc(X) of all exceptional curves on X\in FPMC is finite and generates NE(X). Let \delta_E(X) be the maximum of (-E^2) and p_E(X) the maximum of p_a(E) respectively for E\in Exc(X). For fixed \rho \ge 3, \delta_E and p_E we denote by FPMC_{\rho,\delta_E,p_E} the class of all X\in FPMC such that \rho(X)=\rho, \delta_E(X)=\delta_E and p_E(X)=p_E. We prove that the class FPMC_{\rho,\delta_E,p_E} is bounded: for any X\in FPMC_{\rho,\delta_E,p_E} there exist an ample effective divisor h and a very ample divisor h' such that h^2\le N(\rho,\delta_E) and {h'}^2\le N'(\rho,\delta_E,p_E) where the constants N(\rho,\delta_E)$ and N'(\rho,\delta_E,p_E) depend only on (\rho, \delta_E) and (\rho, \delta_E, p_E) respectively. One can consider Theory of surfaces X\in FPMC as Algebraic Geometry analog of the Theory of arithmetic reflection groups in hyperbolic spaces.