The T-Graph of a Multigraded Hilbert Scheme

Milena Hering, Diane Maclagan

Research output: Contribution to journalArticlepeer-review


The T-graph of a multigraded Hilbert scheme records the zero- and one-dimensional orbits of the T = (K*)(n) action on the Hilbert scheme induced from the T-action on A(n). It has vertices the T-fixed points, and edges the one-dimensional T-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining the scheme parameterizing all one-dimensional torus orbits whose closures contain two given monomial ideals. For this Hilbert scheme we show that the T-graph depends on the ground field, resolving a question of Altmann and Sturmfels.

Original languageEnglish
Pages (from-to)280-297
Number of pages18
JournalExperimental mathematics
Issue number3
Publication statusPublished - Sep 2012


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