## Journal of Differential Geometry

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

### Curve neighborhoods of Schubert varieties

Anders S. Buch and Leonardo C. Mihalcea

#### 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

**Permanent link to this document**

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