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Explicit deformation of a spider algebra to a curvilinear scheme via Möbius generators

David Turturean

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 Möbius 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 Bérczi-Svendsen that every monomial subscheme of $\mathbb{C}^d$ lies in the curvilinear component of the Hilbert scheme of points.

Explicit deformation of a spider algebra to a curvilinear scheme via Möbius generators

Abstract

We construct an explicit flat one-parameter family of 22-dimensional Artinian -algebras whose special fibre is the spider algebra and whose generic fibre is the curvilinear algebra . The construction uses Möbius generators 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 Bérczi-Svendsen that every monomial subscheme of lies in the curvilinear component of the Hilbert scheme of points.
Paper Structure (26 sections, 13 theorems, 52 equations)

This paper contains 26 sections, 13 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Warm-up: the 3-point case
  4. Möbius generators and the basis theorem
  5. The embedding
  6. The basis theorem
  7. The generic ideal
  8. Useful identities
  9. Mixed relations
  10. Pure-power relations
  11. The generic ideal
  12. The weighted Rees family
  13. Weight selection
  14. Homogenisation
  15. Fibres
  16. ...and 11 more sections

Key Result

Theorem 1.1

Every monomial ideal in $\operatorname{Hilb}^n(\mathbb{C}^d)$ lies in the curvilinear component $\operatorname{CHilb}^n_0(\mathbb{C}^d)$.

Theorems & Definitions (31)

  • Theorem 1.1: BercziSvendsen23
  • Theorem 1.2: Main Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 3.1
  • Lemma 4.1: Möbius identity
  • proof
  • Remark 4.2
  • ...and 21 more