The relation between the configuration space and the root systems

1. （与神户大学takayama合著）（3,6）型超几何函数的偏微分方程组存在于射影平面上6点的位形空间上。我们处理了两种构形空间的紧化，研究了这两种空间之间的关系，构造了上述系统的局部基本解。（与M.Yoshida（九州大学）合作）Naruki的交叉比变化被认为是射影平面上6个点的位形空间的紧化。在此基础上，研究了实数投影平面上6条线的位形空间（= 6点的位形空间）。特别地，我们用E_6.3型根系解释了实投影平面上6条线排列的分类。研究了E_7型根的实投影平面7条线的形态与根系的关系。已知在实投影平面上有相互不同的11种7条线排列。我对F_7.4型根系的分类作了解释。（与日本Mukogawa女子大学的T.Fukui合作）我们研究了实投影平面上8条线的形态与F_5型根系的关系。我们的目标是对实投影平面上的8条线排列进行分类，这是一个开放问题。我们用组合论证构造了很多8行排列。

1. (Joint work with T.Takayama (Kobe Univ.)) The system of partial differential equations for the hypergeometric function of type (3,6) lives on the configuration space of 6 points on the projective plane. We treated two kinds of compactifications of the configuration space, studied the relations between these two spaces and constructed local fundamental solutions of the system above.2. (Joint work with M.Yoshida (Kyushu Univ.)) Naruki's cross ratio variety is regarded as a compactification of the configuration space of 6 points on the projective plane. Unsig the properties of this variety, we studied the configuration space of 6 lines (= the configuration space of 6 points) on the real projective plane. In particular, we gave an interpretation of the classification of 6 lines arrangements on the real projective plane in terms of the root system of type E_6.3. I studied the relationship between the configurations of 7 lines the real projective plane and the root system of type E_7. It is known that there are mutually different 11 kinds of 7 lines arrangements on the real projective plane. I gave an interpretation of the classification in terms of the root system of type F_7.4. (Joint work with T.Fukui (Mukogawa Women's Univ.)) We studied the relationship between the configurations of 8 lines the real projective plane and the root system of type F_5. Our aim is the classification of 8 lines arrangements on the real projective plane, which is an open problem. We constructed a lot of 8 lines arrangements by a combinatorial argument.