  We prove a conjecture of Buch and Mihalcea in the case of the incidence
variety X=Fl(1,n-1;n) and determine the structure of its (T-equivariant)
quantum K-theory ring. Our results are an interplay between geometry and
combinatorics. The geometric side concerns Gromov-Witten varieties of 3-pointed
genus 0 stable maps to X with markings sent to Schubert varieties, while on the
combinatorial side are formulas for the (equivariant) quantum K-theory ring of
X. We prove that the Gromov-Witten variety is rationally connected when one of
the defining Schubert varieties is a divisor and another is a point. This
implies that the (equivariant) K-theoretic Gromov-Witten invariants defined by
two Schubert classes and a Schubert divisor class can be computed in the
ordinary (equivariant) K-theory ring of X. We derive a positive Chevalley
formula for the equivariant quantum K-theory ring of X and a positive
Littlewood-Richardson rule for the non-equivariant quantum K-theory ring of X.
The Littlewood-Richardson rule in turn implies that non-empty Gromov-Witten
varieties given by Schubert varieties in general position have arithmetic genus
0.
