Analysis of freeness of hyperplane arrangements and related geometry

2015 Fiscal Year Final Research Report

• Project/Area Number: 24740012
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Kyushu University (2015); Kyoto University (2012-2014)
• Principal Investigator: Abe Takuro 九州大学, マス・フォア・インダストリ研究所, 准教授 (50435971)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 超平面配置 / 自由配置 / 代数学 / 代数幾何学 / ベクトル束 / ルート系とワイル群 / イデアル / 直線配置

Outline of Final Research Achievements

This research program aims at studying the freeness of hyperplane arrangemens more deeply, and investigate its geometry. These aims are achieved and the results are better than we expected. First, for any ideal subarrangements of Weyl arrangements coming from an ideal which is certain subsets of positive roots, we determine its Betti numbers by using freeness which coincides with the dual partition of height distributions of roots in the ideal. Next, we improved Terao's addition-deletion theorem invented in 1980 into the division theorem for free arrangements. By applying it, we enlarged the class in which the freeness depends only on combinatorics, and named divisionally free arrangements. These two results are of the great advances in this research area, so we think this research program is achieved successfully.

Free Research Field

超平面配置に関する数学