Mirror symmetry for certain blowups of Grassmannians

We classify when the blowup of a complex Grassmannian $G(k,n)$ along a smooth Schubert subvariety $Z$ is Fano. We compute almost all the two-point, genus zero Gromov-Witten invariants of the blowup when $Z=G(k,n-1)$. We further prove a mirror symmetry statement for the blowup $X_{2,n}$ of $G(2,n)$ along $G(2,n-1)$, by introducing a toric superpotential $f_{\mathrm{t}\mathrm{o}\mathrm{r}}$ and showing the isomorphism between the Jacobi ring of $f_{\mathrm{t}\mathrm{o}\mathrm{r}}$ and the small quantum cohomology ring $Q{H}^{\ast }(X_{2,n})$.