  We study the arc space of the Grassmannian from the point of view of the
singularities of Schubert varieties. Our main tool is a decomposition of the
arc space of the Grassmannian that resembles the Schubert cell decomposition of
the Grassmannian itself. Just as the combinatorics of Schubert cells is
controlled by partitions, the combinatorics in the arc space is controlled by
plane partitions (sometimes also called 3d partitions). A combination of a
geometric analysis of the pieces in the decomposition and a combinatorial
analysis of plane partitions leads to invariants of the singularities. As an
application we reduce the computation of log canonical thresholds of pairs
involving Schubert varieties to an easy linear programming problem. We also
study the Nash problem for Schubert varieties, showing that the Nash map is
always bijective in this case.
