Curve neighborhoods and minimal degrees in quantum products

TL;DR

This paper establishes the existence and uniqueness of a minimal degree in quantum products of point classes on homogeneous spaces, providing an explicit formula and geometric construction for this degree.

Contribution

It introduces a formula for the minimal degree in quantum products on homogeneous spaces and constructs explicit curves realizing this degree.

Findings

Existence and uniqueness of the minimal degree d_X in quantum point products.

Explicit formula for d_X using cascade of orthogonal roots.

Construction of an explicit curve of degree d_X passing through two general points.

Abstract

Let $G$ be a connected, simply connected, simple, complex, linear algebraic group. Let $P$ be an arbitrary parabolic subgroup of $G$. Let $X=G/P$ be the $G$-homogeneous projective space attached to this situation. We consider the (small) quantum cohomology ring $(QH^*(X),\star)$ attached to $X$. We prove that there exists a unique degree $d$ which is minimal with the property that $q^d$ occurs with non-zero coefficient in the quantum product of two point classes. We denote this minimal degree in $\mathrm{pt}\star\mathrm{pt}$ by $d_X$. We give an explicit formula to compute $d_X$ in terms of the cascade of orthogonal roots. We construct an explicit curve of degree $d_X$ passing through two general points in $X$. Moreover, we prove that $d_X$ is the unique maximal element of the set of all minimal degrees in some quantum product of two Schubert classes.