We show that the Schubert calculus of enumerative geometry is real, for
special Schubert conditions. That is, for any such enumerative problem, there
exist real conditions for which all the a priori complex solutions are real.Comment: 5 page

Sottile, Frank

English

arXiv

. We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real. Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures [4]. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real [9]. Similarly, given any problem of enumerating lines in projective space incident on some general fixed linear subspaces, there are real fixed subspaces such that each of the (finitely many) incident lines are real [12]. Other examples are shown in [11, 13], and the case of 462 4-planes meeting 12 general 3-planes in R  7  is due to an heroic symbolic computation [3]. For any problem of enumerating p-planes having excess int..

The special Schubert calculus is real

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