On almost revlex ideals with Hilbert function of complete intersections · arXivDesk
1803.02330Mar 6, 2018Title changed. Generalization and improvements of the results of the previous version. New sections concerning bounds on the dimension of the tangent space to a Hilbert scheme at a quasi-stable ideal, and sufficient conditions for an almost revlex ideal to be a singularity in a Hilbert scheme
On almost revlex ideals with Hilbert function of complete intersections
In this paper, we investigate the behavior of almost reverse lexicographic ideals with the Hilbert function of a complete intersection. More precisely, over a field K, we give a new constructive proof of the existence of the almost revlex ideal J⊂K[x1,…,xn]
Nearby in the stack
, with the same Hilbert function as a complete intersection defined by
n
forms of degrees
d1≤⋯≤dn
. Properties of the reduction numbers for an almost revlex ideal have an important role in our inductive and constructive proof, which is different from the more general construction given by Pardue in 2010. We also detect several cases in which an almost revlex ideal having the same Hilbert function as a complete intersection corresponds to a singular point in a Hilbert scheme. This second result is the outcome of a more general study of lower bounds for the dimension of the tangent space to a Hilbert scheme at stable ideals, in terms of the number of minimal generators.