Some real and unreal enumerative geometry for flag manifolds

Details Some real and unreal enumerative geometry for flag manifolds Some real and unreal enumerative geometry for flag manifolds

2000-02-24

arxiv.org/abs/math/0002207v2

Michigan Math Journal, 48 (2000) pp. 573-592.

Mathematics

Algebraic Geometry

19 pages, LaTeX-2e; Updated and final version. To appear in the issue of Michigan Mathematical Journal dedicated to Bill Fulto

Scientific paper

We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative geometric problems on flag manifolds may have all their solutions be real and modify this method to show that another class may have no real solutions, which is a new phenomenon. This method originated in a numerical homotopy continuation algorithm adapted to the special Schubert calculus on Grassmannians and in principle gives optimal numerical homotopy algorithms for finding explicit solutions to these other enumerative problems.

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