Rigid and non-smoothable Schubert classes

Abstract

A Schubert class $\sigma_\lambda$ in the Grassmannian $G(k, n)$ is rigid if the only proper subvarieties representing that class are Schubert varieties $\Sigma_\lambda$. We prove that a Schubert class $\sigma_\lambda$ is rigid if and only if it is defined by a partition $\lambda$ satisfying a simple numerical criterion. If $\lambda$ fails this criterion, then there is a corresponding hyperplane class in another Grassmannian $G(k', n')$ such that the deformations of the hyperplane in $G(k', n')$ yield non-trivial deformations of the Schubert variety $\Sigma_\lambda$. We also prove that, if a partition λ contains a sub-partition defining a rigid and singular Schubert class in another Grassmannian $G(k', n')$, then there does not exist a smooth subvariety of $G(k, n)$ representing $\sigma_\lambda$.