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Zbl 1213.14117
Hillar, Christopher; Garc\'ia-Puente, Luis; del Campo, Abraham Mart\'in; Ruffo, James; Teitler, Zach; Johnson, Stephen L.; Sottile, Frank
Experimentation at the frontiers of reality in Schubert calculus. (English)
[A] Amdeberhan, Tewodros (ed.) et al., Gems in experimental mathematics. AMS special session on experimental mathematics, Washington, DC, January 5, 2009. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 517, 365-380 (2010). ISBN 978-0-8218-4869-2/pbk

The Schubert calculus asks for the linear spaces that have specified positions with respect to other, fixed (flags of) linear spaces. The specified positions are a Schubert problem. The fixed linear spaces imposing the conditions give an instance of the Schubert problem. The number of solutions depends on the Schubert problem, while the solutions depend on the instance. The conjecture of Boris Shapiro and Michael Shapiro says that if the fixed linear space osculates a rational normal curve, then all solutions to the Schubert problem are real. The paper gives an overview about the state of the art concerning the conjecture. Based on ideas obtained by a lot of computer experiments the Shapiro conjecture was proved for Grassmannians [{\it E. Mukhin, V. Tarasob, A. Varchenko}, Ann. Math. (2) 170, No. 2, 863--881 (2009; Zbl 1213.14101)]. \par Computer experiments are described to test modified conjectures. One of them is the so--called secant conjecture. A flag is a secant along an interval $I$ of a curve if every subspace in the flag is spanned by its intersections with $I$. The secant conjecture asserts that if the flags in a Schubert problem on a Grassmannian are disjoint in that they are secant along disjoint intervals of a rational normal curve, then every solution is real. A computer experiment is described verifying the secant conjecture in $4\, 568\, 553$ instances using $4473$ GigaHertz--years.
[Gerhard Pfister (Kaiserslautern)]

MSC 2000:: *14Q15 Higher dimensional varieties
14N15 Classical problems in algebraic geometry

Keywords: Schubert calculus; Shapiro conjecture; secant conjecture; {\tt Singular}

Citations: Zbl 1213.14101

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