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 $\mathbb{c}$ with finite polyhedral Mori cone $NE(X)$ $\subset NS(X)\otimes \mathbb{R}$. 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 $(-C^2)$ and $p_E(X)$ the maximum of $p_a(C)$ respectively for all $C\in \,$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 algebraic surfaces $X\in FPMC$ 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 in the following sense: 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.