Rational elliptic surfaces and the trigonometry of tetrahedra

• 成果类型: Article
• 署名作者: Rudenko, Daniil
• 署名单位: University of Chicago
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01066-w
• 发表日期: 2022
• 页码: 211-246
• 关键词: VOLUME; MODULI

摘要

We study the trigonometry of non-Euclidean tetrahedra using tools from algebraic geometry. We establish a bijection between non-Euclidean tetrahedra and certain rational elliptic surfaces. We interpret the edge lengths and the dihedral angles of a tetrahedron as values of period maps for the corresponding surface. As a corollary we show that the cross-ratio of the exponents of the solid angles of a tetrahedron is equal to the cross-ratio of the exponents of the perimeters of its faces. The Regge symmetries of a tetrahedron are related to the action of the Weyl group W(D-6) on the Picard lattice of the corresponding surface.