## Journal of Differential Geometry

### Curve neighborhoods of Schubert varieties

#### Abstract

A previous result of the authors with Chaput and Perrin 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.

#### Article information

Source
J. Differential Geom., Volume 99, Number 2 (2015), 255-283.

Dates
First available in Project Euclid: 16 January 2015

https://projecteuclid.org/euclid.jdg/1421415563

Digital Object Identifier
doi:10.4310/jdg/1421415563

Mathematical Reviews number (MathSciNet)
MR3302040

Zentralblatt MATH identifier
06423472

#### Citation

Buch, Anders S.; Mihalcea, Leonardo C. Curve neighborhoods of Schubert varieties. J. Differential Geom. 99 (2015), no. 2, 255--283. doi:10.4310/jdg/1421415563. https://projecteuclid.org/euclid.jdg/1421415563