Explicit deformation of a spider algebra to a curvilinear scheme via M\"obius generators

TL;DR

This paper constructs an explicit flat family of 22-dimensional Artinian algebras connecting a specific spider algebra to a curvilinear algebra, illustrating a general geometric phenomenon about monomial subschemes in Hilbert schemes.

Contribution

It provides the first explicit construction of a flat family linking a spider algebra to a curvilinear algebra using M"obius generators and Rees degeneration.

Findings

Explicit family connecting spider and curvilinear algebras

Uses M"obius generators and divided-difference coordinates

Witnesses the inclusion of monomial subschemes in the curvilinear component

Abstract

We construct an explicit flat one-parameter family of 22-dimensional Artinian $k$-algebras whose special fibre is the spider algebra $k[x,y,z]/(x^8, y^8, z^8, xy, xz, yz)$ and whose generic fibre is the curvilinear algebra $k[t]/(t^{22})$. The construction uses M\"obius generators $u_a = t/(1-at)$ inside the curvilinear ring together with a divided-difference change of coordinates, and produces the family via a weighted Rees degeneration with integer coefficients. This gives an explicit one-parameter family witnessing, for this spider ideal, the general phenomenon proved by B\'erczi-Svendsen that every monomial subscheme of $\mathbb{C}^d$ lies in the curvilinear component of the Hilbert scheme of points.