Translator Disclaimer
2000 A remark on algebraic surfaces with polyhedral Mori cone
Viacheslav V. Nikulin
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 $NE(X)$ $\subset NS(X)\otimes \mathbb{R}$. 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.


Download Citation

Viacheslav V. Nikulin. "A remark on algebraic surfaces with polyhedral Mori cone." Nagoya Math. J. 157 73 - 92, 2000.


Published: 2000
First available in Project Euclid: 27 April 2005

zbMATH: 0958.14026
MathSciNet: MR1752476

Primary: 14J26
Secondary: 14C22

Rights: Copyright © 2000 Editorial Board, Nagoya Mathematical Journal


Vol.157 • 2000
Back to Top