Real solutions of a problem in enumerative geometry
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- 作者：László M. Fehér ; Ákos K. Matszangosz
- 关键词：Real enumerative geometry ; Enumerative real algebraic geometry ; Modulo four congruence ; Real Grassmannian ; Real Schubert variety ; Four medials
- 刊名：Periodica Mathematica Hungarica
- 出版年：2016
- 出版时间：December 2016
- 年：2016
- 卷：73
- 期：2
- 页码：137-156
- 全文大小：544 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Sciences
Mathematics - 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
- ISSN：1588-2829
- 卷排序：73
文摘
We study a 2-parameter family of enumerative problems over the reals. Over the complex field, these problems can be solved by Schubert calculus. In the real case the number of solutions can be different on the distinct connected components of the configuration space, resulting in a solution function. The cohomology calculation in the real case only gives the signed sum of the solutions, therefore in general it only gives a lower bound on the range of the solution function. We calculate the solution function for the 2-parameter family and we show that in the even cases the solution function is constant modulo 4. We show how to determine the sign of a solution and describe the connected components of the configuration space. We translate the problem to the language of quivers and also give a geometric interpretation of the sign. Finally, we discuss what aspects might be considered when solving other real enumerative problems.