A toric degeneration of Kronecker moduli spaces

TL;DR

This paper constructs a toric degeneration of Kronecker moduli spaces using SAGBI bases, generalizing known degenerations of Grassmannians, and describes the associated moment polytope as an intersection of Gelfand--Cetlin polytopes.

Contribution

It introduces a finite SAGBI basis for the coordinate ring of Kronecker quiver moduli spaces, leading to a new toric degeneration generalizing Grassmannian degenerations.

Findings

Established a SAGBI basis indexed by primitive semi-standard tableaux pairs.

Derived a toric degeneration to a normal toric variety.

Described the moment polytope as an intersection of two Gelfand--Cetlin polytopes.

Abstract

In this paper, we show that there is a finite SAGBI basis of the coordinate ring of a Kronecker quiver moduli space, indexed by primitive semi-standard tableaux pairs. This induces a toric degeneration of the Kronecker moduli space to a normal toric variety, a generalization of the toric degeneration of the Grassmannian to the Gelfand--Cetlin polytope constructed by Gonciulea--Lakshmibai. The moment polytope of the degenerate toric variety can be described as the intersection of two Gelfand--Cetlin polytopes. We explain when this can be generalized to degenerations coming from matching fields.