Curve neighborhoods of Schubert varieties. (English) Zbl 1453.14117

Summary: A previous result of the authors with P.-E. Chaput and N. Perrin [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 3, 477–494 (2013; Zbl 1282.14016)] states that the closure of the union of all rational curves of fixed degree passing through a Schubert variety in a homogeneous space \(G/P\) is again a Schubert variety. In this paper, we identify this Schubert variety explicitly in terms of the Hecke product of Weyl group elements. We apply our result to give an explicit formula for any two-point Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula and its equivariant generalization. We also recover a formula for the minimal degree of a rational curve between two given points in a cominuscule variety.


14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)


Zbl 1282.14016
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