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Experimentation and conjectures in the real Schubert calculus for flag manifolds. (English) Zbl 1111.14049

The Shapiro conjecture for Grassmannians states that the zero-dimensional intersection of Schubert subvarieties of a Grassmannian consists entirely of real points if the Schubert varieties are given by flags osculating a real rational normal curve. It was proven for Grassmannians of codimension 2 subspaces by A. Eremenko and A. Gabrielov [Ann. Math. (2) 155, No. 1, 105–129 (2002; Zbl 0997.14015)] and in full generality by E. Mukhin, V. Tarasov and A. Varchenko [Ann. Math. (2) 170, No. 2, 863–881 (2009; Zbl 1213.14101), preprint arXiv:math.AG/0512299]. A similar conjecture for the flag varieties fails as shown by F. Sottile [Exp. Math. 9, 161–182 (2000; Zbl 0997.14016)].
In the present article, the authors suggest a refinement of the Shapiro conjecture for flag manifolds by including some monotonicity conditions, and they confirm this refinement in a massive computational experiment and in a number of particular cases.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14N15 Classical problems, Schubert calculus
14P99 Real algebraic and real-analytic geometry

Software:

SINGULAR
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