A Congruence Modulo Four for Real Schubert Calculus with Isotropic Flags

Abstract We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.

[1]  Steven L. Kleiman,et al.  The transversality of a general translate , 1974 .

[2]  Joe Harris,et al.  Divisors on general curves and cuspidal rational curves , 1983 .

[3]  Chiaki Itoh,et al.  Where Is the Σb , 1992 .

[4]  Frank Sottile,et al.  The special Schubert calculus is real , 1999 .

[5]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[6]  Andrei Gabrielov,et al.  Degrees of Real Wronski Maps , 2002, Discret. Comput. Geom..

[7]  A. Eremenko,et al.  Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry , 2004, math/0405196.

[8]  Eugenii Shustin,et al.  Welschinger invariant and enumeration of real rational curves , 2003 .

[9]  Jean-Yves Welschinger,et al.  Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry , 2003 .

[10]  Frank Sottile,et al.  Lower bounds for real solutions to sparse polynomial systems , 2004 .

[11]  Eugenii Shustin,et al.  Logarithmic equivalence of Welschinger and Gromov-Witten invariants , 2004 .

[12]  E. Mukhin,et al.  The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz , 2005 .

[13]  E. Mukhin,et al.  Schubert calculus and representations of the general linear group , 2007, 0711.4079.

[14]  Frank Sottile.  General isotropic flags are general (for Grassmannian Schubert calculus) , 2008 .

[15]  Kevin Purbhoo.  Reality and transversality for Schubert calculus in OG (n, 2n+1) , 2009 .

[16]  C. Okonek,et al.  Intrinsic signs and lower bounds in real algebraic geometry , 2011, 1112.3851.

[17]  Frank Sottile,et al.  A congruence modulo four in real Schubert calculus , 2012 .

[18]  S.Finashin,et al.  Abundance of real lines on real projective hypersurfaces , 2012, 1201.2897.

[19]  C. Okonek,et al.  A wall-crossing formula for degrees of Real central projections , 2012, 1206.4271.

[20]  Frank Sottile,et al.  Lower Bounds in Real Schubert Calculus , 2013, 1308.4381.

[21]  Frank Sottile,et al.  Experimentation in the Schubert Calculus , 2013 .

[22]  E. Mukhin,et al.  Lower bounds for numbers of real solutions in problems of Schubert calculus , 2014 .

[23]  Brian D. O. Anderson,et al.  Counting Critical Formations on a Line , 2014, SIAM J. Control. Optim..

[24]  László M. Fehér,et al.  Real solutions of a problem in enumerative geometry , 2014, Periodica Mathematica Hungarica.