Smoothing effective linear combinations of Schubert classes
In a homogeneous variety $G/P$, classes $[X_w]$ of Schubert varieties $X_w$ generate the homology of $G/P$. In 1961, Borel and Haefliger posed the question: when does there exist a subvariety $Y$ in $G/P$ that is homologous to $X_w$? Many authors have studied this question, particularly in cominiscule homogeneous varieties and the special case of finding smooth subvarieties that represent Schubert classes. This leads to my question: In $G/P$, which effective linear combinations of Schubert classes can be represented by an effective linear combination of smooth subvarieties? I will discuss a strategy for answering this and give explicit examples, which includes showing how to compute the Schubert expansion of a homogeneous subvariety by using divided difference operators.

