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Newton-Okounkov bodies for nested Hilbert schemes

Ian Cavey, Eugene Gorsky, Alexei Oblomkov, Joshua P. Turner

TL;DR

This work computes explicit structures for line bundles on the nested Hilbert scheme $\mathrm{Hilb}^{n,n+1}(\mathbb{C}^2)$ by relating global sections to Haiman's ideals $J$ and $I$ and using trailing-term bases. It establishes a birational bridge with $\mathrm{Hilb}^n(\mathrm{Bl}_0\mathbb{C}^2)$, enabling a transfer of sections and a compact description via $A^{m+k}[x,y] \cap I^k$, and it provides precise trailing-term combinatorics culminating in Newton-Okounkov bodies. The main contributions are (i) a sharp description of $H^0(\mathrm{Hilb}^{n,n+1}(\mathbb{C}^2), \mathcal{O}(m,k))$ in terms of $J^{m+k}\cap I^k$, (ii) a birational equivalence with the blow-up Hilbert scheme that clarifies cohomology and Picard data, and (iii) an explicit characterization of trailing terms, Hilbert series, and Newton-Okounkov bodies, revealing deep connections to wall-crossing and representation-theoretic combinatorics.

Abstract

We study sections of line bundles on the nested Hilbert scheme of points on the affine plane. We describe the spaces of sections in terms of certain ideals introduced by Haiman, and find explicit bases for them by analyzing the trailing terms in some monomial order. As a consequence, we compute the Newton-Okounkov bodies for nested Hilbert schemes.

Newton-Okounkov bodies for nested Hilbert schemes

TL;DR

This work computes explicit structures for line bundles on the nested Hilbert scheme by relating global sections to Haiman's ideals and and using trailing-term bases. It establishes a birational bridge with , enabling a transfer of sections and a compact description via , and it provides precise trailing-term combinatorics culminating in Newton-Okounkov bodies. The main contributions are (i) a sharp description of in terms of , (ii) a birational equivalence with the blow-up Hilbert scheme that clarifies cohomology and Picard data, and (iii) an explicit characterization of trailing terms, Hilbert series, and Newton-Okounkov bodies, revealing deep connections to wall-crossing and representation-theoretic combinatorics.

Abstract

We study sections of line bundles on the nested Hilbert scheme of points on the affine plane. We describe the spaces of sections in terms of certain ideals introduced by Haiman, and find explicit bases for them by analyzing the trailing terms in some monomial order. As a consequence, we compute the Newton-Okounkov bodies for nested Hilbert schemes.

Paper Structure

This paper contains 10 sections, 30 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Birational relation
  3. Nested Hilbert schemes
  4. Birational relation
  5. Localization formulas
  6. Global Sections
  7. Trailing Terms of Global Sections
  8. Trailing Terms Analysis
  9. Character Formulas
  10. Newton-Okounkov Bodies

Key Result

Theorem 1.1

For $m,k\ge 0$ the global sections of $\mathcal{O}(m,k)$ are identified with the $\mathrm{sgn}(m+k)$-component of $J^{m+k}\cap I^k\subseteq \mathbb{C}[x_1,y_1,\dots,x_n,y_n,x,y]$, where the ideal $J$ is given by eq: def J intro and

Figures (4)

  • Figure 1: Quadruples of points satisfying $S=S^{(1)}+S^{(2)}$, so that $S$ is the trailing term exponent of an element of $A^2$.
  • Figure 2: The decompositions of $\ell_1(S)$ and $\ell_2(S)$ obtained by modifying the decomposition $S = S^{(1)}+S^{(2)}$ from Example \ref{['ex:c2pointdecomp']}.
  • Figure 3: The vertices of the cones appearing appearing as subsets of $\mathcal{P}(2,1)$ in the case $a_1=0$, where the four dimensional vector $(a_1,a_2,b_1,b_2)$ is represented as a pair of points $p_1 = (a_1,b_1)$ and $p_2 =(a_2,b_2)$.
  • Figure 4: The vertices of the cones appearing appearing as subsets of $\mathcal{P}(2,1)$ in the case $a_1\geq 1$, where the four dimensional vector $(a_1,a_2,b_1,b_2)$ is represented as a pair of points $p_1 = (a_1,b_1)$ and $p_2 =(a_2,b_2)$.

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 53 more