Rigid and Non-smoothable Schubert Classes

A Schubert class in the Grassmannian is rigid if the only proper subvarieties representing that class are Schubert varieties. The hyperplane class σ1 is not rigid because a codimension one Schubert cycle can be deformed to a smooth hyperplane section. In this paper, we show that this phenomenon accounts for the failure of rigidity in Schubert classes. More precisely, we prove that a Schubert class in G(k, n) is not rigid if and only if the partition λ = (λ1, . . . , λk) defining the class has a part λi such that n− k ≥ λi−1 > λi and λi = λi+1 + 1. Under these assumptions on λ, the parts λi and λi+1 determine a partial flag isomorphic to one defining a hyperplane class in another Grassmannian G(k′, n′). Using a deformation of the hyperplane in G(k′, n′), we can deform the Schubert cycle Σλ. Otherwise, the Schubert class σλ is rigid. We also show that if the partition λ contains the partition defining a rigid and singular Schubert cycle in some Grassmannian as a sub-partition, then σλ cannot be represented by a smooth subvariety of G(k, n). More precisely, if λ does not have the form λ1 = · · · = λj1 = n− k, λi = λi+1 + 1 for j1 < i < j2 and λi = λj2 for i ≥ j2, then the Schubert class σλ cannot be represented by a smooth subvariety of G(k, n).