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

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.

Small elementary components of Hilbert schemes of points

“…Our examples, however, were arrived at by different methods, and can be best understood in terms of the T-graph of the Hilbert scheme Hilb 𝑑 (A 𝑛 ); this is the graph whose vertices correspond to torus-fixed points of Hilb 𝑑 (A 𝑛 ), and whose edges correspond to torus-invariant curves linking these fixed points. The T-graph was defined in [2] and studied further in [19,32,35], mostly in dimension 𝑛 = 2. To produce our examples, we constructed edges of the T-graph corresponding to curves in Hilb 𝑑 (A 4 ) whose general point is a smooth point with small tangent space dimension.…”