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Hilbert scheme of points on non-reduced nodal curves

Yuze Luan

Abstract

We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, $x^uy^v=0$. Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on introducing filtrations on ideals, together with a valuation adapted to the non-reduced structure, which allows us to analyze generators and their degrees of freedom in a systematic way. In particular, each stratum is affine when $u=1,2$; and each stratum is isomorphic to an algebraic torus times an affine space, $(\mathbb{C}^*)^{m_1} \times \mathbb{C}^{m_2}$, when $u=v,v-1,v-2$. We consequently compute the Poincaré polynomials of the punctual Hilbert scheme of points on curves $x^uy^v=0$ when $u=1,2,v-2,v-1,v$. As an application, we prove the colored Oblomkov-Rasmussen-Shende conjecture for the Hopf link for $u=1, v$ arbitrary, showing that the Poincaré polynomial is the row-colored link homology up to change of variables.

Hilbert scheme of points on non-reduced nodal curves

Abstract

We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, . Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on introducing filtrations on ideals, together with a valuation adapted to the non-reduced structure, which allows us to analyze generators and their degrees of freedom in a systematic way. In particular, each stratum is affine when ; and each stratum is isomorphic to an algebraic torus times an affine space, , when . We consequently compute the Poincaré polynomials of the punctual Hilbert scheme of points on curves when . As an application, we prove the colored Oblomkov-Rasmussen-Shende conjecture for the Hopf link for arbitrary, showing that the Poincaré polynomial is the row-colored link homology up to change of variables.

Paper Structure

This paper contains 43 sections, 37 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. The ORS conjecture
  3. Verified cases of the ORS conjecture.
  4. Affine paving of the Hilbert scheme
  5. Methods of this paper
  6. Results of this paper
  7. Affine paving of $\mathrm{Hilb}^n(\{xy^v=0\},0)$ and the ORS conjecture.
  8. Affine paving of $\mathrm{Hilb}^{n}(\{x^2y^v=0\},0)$.
  9. Torus-like Stratification for $u=v, v-1$ or $v-2$.
  10. Virtual Poincaré polynomial of the Hilbert scheme $\mathrm{Hilb}^n(\{x^uy^v=0\},0)$ for $u=v,v-1, v-2$.
  11. New stratification of the punctual Hilbert scheme of points on $\mathbb{C}^2$.
  12. Predictions on the colored knot homology
  13. Khovanov-Rozansky knot homology of an algebraic link
  14. Hilbert scheme of points and virtual Poincaré polynomial
  15. Reduced nodal singularity
  16. ...and 28 more sections

Key Result

Theorem 1.2

The Hilbert scheme $\mathrm{Hilb}^{n}(\{xy^v=0\},0)$ has an affine stratification, and each stratum is indexed by a partition $(i_0, ..., i_{N})$ such that $i_k \leq 1$ for all $k \geq v$. The dimension of the stratum is the number of boxes from the $1$st row to the $v$th row, i.e. $\sum_{k=1}^{v}i_ $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (91)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3: Colored ORS conjecture for the Hopf link
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 2.1
  • ...and 81 more