The spine of the T-graph of the Hilbert scheme of points in the plane

TL;DR

This paper constructs a field-independent subgraph called the spine within the T-graph of the Hilbert scheme of points in the plane, providing insights into tropical ideals and stratification.

Contribution

It introduces a novel, field-independent subgraph of the T-graph, called the spine, and describes tropical ideals for certain edges, enhancing understanding of the Hilbert scheme's structure.

Findings

The spine is independent of the underlying field.

Certain edges in the spine have associated tropical ideals.

The work refines the tropical stratification of the Hilbert scheme.

Abstract

The torus T of projective space also acts on the Hilbert scheme of subschemes of projective space. The T-graph of the Hilbert scheme has vertices the fixed points of this action, and edges connecting pairs of fixed points in the closure of a one-dimensional orbit. In general this graph depends on the underlying field. We construct a subgraph, which we call the spine, of the T-graph of Hilb^m(A^2) that is independent of the choice of infinite field. For certain edges in the spine we also give a description of the tropical ideal, in the sense of tropical scheme theory, of a general ideal in the edge. This gives a more refined understanding of these edges, and of the tropical stratification of the Hilbert scheme.