关于秩为3的双曲根系的分类。部件III
On the classification of hyperbolic root systems of the rank three. Part III
摘要：
参见alg-geom/9711032和alg-geom/9712033中的第I部分和第II部分。本文对具有限制算术型的秩为3的极大双曲根系和广义格Weyl向量$ρ$进行了分类，其中$ρ^2<0$（i. e.双曲线型）。给出了秩为3的双曲型基本双曲格的分类。对于椭圆型（当$ρ^2>0$）和抛物线型（当$ρ^2=0$），在第一部分和第二部分中完成。我们应用与椭圆和抛物型相同的参数：双曲空间中多面体的窄部方法，以及中心对称的类数。但我们应该说，对于双曲型，所有的考虑都要复杂得多，需要更多的计算和时间。这些结果是重要的，例如，理论的洛伦兹卡茨-穆迪代数和某些方面的镜像对称。我们也应用这些结果来证明几乎有限多面体Mori锥的代数曲面族的有界性（见数学AG/9806047关于这个问题）。
英文摘要：
See Parts I and II in alg-geom/9711032 and alg-geom/9712033. Here we classify maximal hyperbolic root systems of the rank three having restricted arithmetic type and a generalized lattice Weyl vector $ρ$ with $ρ^2<0$ (i. e. of the hyperbolic type). We give classification of all reflective of hyperbolic type elementary hyperbolic lattices of the rank three. For elliptic (when $ρ^2>0$) and parabolic (when $ρ^2=0$) types it was done in Parts I and II. We apply the same arguments as for elliptic and parabolic types: the method of narrow parts of polyhedra in hyperbolic spaces, and class numbers of central symmetries. But we should say that for the hyperbolic type all considerations are much more complicated and required much more calculations and time. These results are important, for example, for Theory of Lorentzian Kac--Moody algebras and some aspects of Mirror Symmetry. We also apply these results to prove boundedness of families of algebraic surfaces with almost finite polyhedral Mori cone (see math.AG/9806047 about this subject).
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