1999/05/24 by Viacheslav V. Nikulin, Nikulin, Viacheslav V.
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math.AG #math.QA
paper · pdf · doi:10.48550/arxiv.math/9905150
102 pages, AMS-Tex
arxiv created 1999/05/24 · openalex publication_date 1999/05/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ρ2<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 ρ2>0) and parabolic (when ρ2=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).