The arc space of the Grassmannian

Abstract

Abstract. 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.

The Nash problem for torus actions of complexity one

“…Following [Ish08,DN17,FdBPPPP17], what we call the generalized Nash problem asks for a meaningful interpretation of the Nash order on DV(X). The problem is well-understood for equivariant valuations on toric varieties ( [Ish08], see also 2.28 below) and determinantal varieties ( [Doc13]).…”

“…The problem is well-understood for equivariant valuations on toric varieties ( [Ish08], see also 2.28 below) and determinantal varieties ( [Doc13]). In [DN17], some partials results are obtained concerning the generalized Nash problem for contact strata in Grassmanians (See also [Mou14,MP18,KMPT20] for a variant of this problem, namely the embedded Nash problem, for a class of surface singularities.). Theorem 5.9 below solves the generalized Nash problem for equivariant valuations on non-rational normal varieties equipped with a complexity one torus action.…”

“…The first example is again due to Ishii and Kollar in [IK03]: they showed that the Nash problem holds for toric singularities; subsequent refinements of this result are to be found in [Ish05]. Nowadays, in dimension 3, the Nash problem is known to hold for the following families: some quasi-rational hypersurface singularities ( [LA11]), some families of 3-dimensional hypersurfaces ( [LA16]), possibly reducible quasi-ordinary hypersurface singularities ( [GP07]), a certain class of normal isolated singularities of arbitrary dimension, ( [PPP08]), Schubert varieties ( [DN17]). Being given two divisorial valuations ν, ν ′ on an algebraic variety, the condition: "any arc with order ν ′ is a limit of arcs with order ν" defines a poset structure on the set of divisorial valuations, the Nash order, and the Nash valuations may be understood as the minimal elements for the Nash order of the set of valuations centered at the singular locus.…”

“…The generalized Nash problem asks for a meaningful description of the Nash order. It has been solved for equivariant valuations on toric varieties ( [Ish04,Ish08]) and determinantal varieties ( [Doc13]); in [DN17], some partial results are obtained for Grassmanians. In its full generality, it seems to be a very difficult problem, even for valuations on the affine plane (see [FdBPPPP17]).…”