Nikulin, Viacheslav V. A remark on algebraic surfaces with polyhedral Mori cone. (English) Zbl 0958.14026 Nagoya Math. J. 157, 73-92 (2000). We denote by FPMC the class of all non-singular projective algebraic surfaces \(X\) over \(\mathbb{C}\) with finite polyhedral Mori cone \(\text{NE}(X)\subset \text{NS}(X)\otimes \mathbb{R}\). If \(\rho(X)= \text{rk NS}(X)\geq 3\), then the set \(\text{Exc} (X)\) of all exceptional curves on \(X\in\text{FPMC}\) is finite and generates \(\text{NE}(X)\). Let \(\delta_E(X)\) be the maximum of \((-C^2)\) and \(p_E(X)\) the maximum of \(p_a(C)\), respectively for all \(C\in\text{Exc}(X)\). For fixed \(\rho\geq 3\), \(\delta_E\) and \(p_E\) we denote by \(\text{FPMC}_{\rho, \delta_E, p_E}\) the class of all algebraic surfaces \(X\in\text{FPMC}\) such that \(\rho(X) =\rho\), \(\delta_E(X) =\delta_E\) and \(p_E(X)=p_E\). We prove that the class \(\text{FPMC}_{\rho, \delta_E,p_E}\) is bounded in the following sense: For any \(X\in\text{FPMC}_{\rho, \delta_E,p_E}\) there exist an ample effective divisor \(h\) and a very ample divisor \(h'\) such that \(h^2\leq N (\rho, \delta_E)\) and \(h^{\prime 2}\leq N'(\rho, \delta_E,p_E)\) 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 the theory of surfaces \(X\in\text{FPMC}\) as an algebraic geometry analog of the theory of arithmetic reflection groups in hyperbolic spaces. 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