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arXiv · 2111.00819

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

Diane Maclagan·Rob Silversmith
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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.

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Diane Maclagan, Rob Silversmith. 2021-11-01. The spine of the T-graph of the Hilbert scheme of points in the plane. https://arxiv.org/abs/2111.00819

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