Counting Dimensions of Tangent Spaces to Hilbert Schemes of Points
Abstract
In this paper we prove two results which further classify smoothness properties of Hilbert schemes of points. This is done by counting classes of arrows on Young diagrams corresponding to monomial ideals, building on the approach taken by Jan Cheah to show smoothness in the 2 dimensional case. We prove sufficient conditions for points to be smooth on Hilbert schemes of points in 3 dimensions and on nested schemes in 2 dimensions in terms of the geometry of the Young diagram. In particular, we proved that when the region between the two diagrams at a point of the nested scheme is rectangular, the corresponding point is smooth. We also proved that if the three dimensional Young diagram at a point can be oriented such that its horizontal layers are rectangular, then the point is smooth.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.09841
 arXiv:
 arXiv:2010.09841
 Bibcode:
 2020arXiv201009841M
 Keywords:

 Mathematics  Algebraic Geometry;
 14C05;
 05E10
 EPrint:
 14 pages, 14 figures