Real rational curves in Grassmannians

Sottile, Frank

doi:10.1090/S0894-0347-99-00323-9

Real rational curves in Grassmannians
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by Frank Sottile

J. Amer. Math. Soc. 13 (2000), 333-341

DOI: https://doi.org/10.1090/S0894-0347-99-00323-9

Published electronically: October 25, 1999

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Abstract:

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 given general figures. For the problem of plane conics tangent to five general (real) conics, the surprising answer is that all 3264 may be real. Similarly, given any problem of enumerating $p$-planes incident on some given general subspaces, there are general real subspaces such that each of the (finitely many) incident $p$-planes is real. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions real.

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