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Nef cones of Hilbert schemes of points on some K3 surfaces

Uttaran Dutta, Sean Edwards, Neelarnab Raha

TL;DR

The paper develops a unified framework to compute nef cones of Hilbert schemes of points on K3 surfaces by combining Bridgeland stability techniques with classical geometric methods, focusing on Mori dream K3 surfaces of Picard rank 2. It identifies a central nef-cone candidate Λ, shows its nefness for large n under explicit Pell-type and polynomial bounds across three case distinctions, and provides tight small-n bounds and a direct computation for the notable case X^{[2]} when k=2. The results extend to nested Hilbert schemes X^{[n,n+1]} for large n, illustrating how pullbacks and diff-divisors control positivity. The work highlights how stability-wall analysis (Bayer–Macrì) and Mori-theoretic considerations yield precise birational-geometric information for Hilbert and nested Hilbert schemes on K3 surfaces, with potential generalizations to other surface classes.

Abstract

We illustrate the typical usage of Bayer and Macrì's Positivity Lemma to compute the nef cones of the Hilbert schemes $X^{[n]}$ by combining the Bridgeland stability methods (for large $n$) and classical methods (for small $n$). We use Mori dream K3 surfaces $X$ of Picard rank $2$ as our working example. We also compute the nef cones of the nested Hilbert schemes $X^{[n,n+1]}$ for such $X$, for large $n$.

Nef cones of Hilbert schemes of points on some K3 surfaces

TL;DR

The paper develops a unified framework to compute nef cones of Hilbert schemes of points on K3 surfaces by combining Bridgeland stability techniques with classical geometric methods, focusing on Mori dream K3 surfaces of Picard rank 2. It identifies a central nef-cone candidate Λ, shows its nefness for large n under explicit Pell-type and polynomial bounds across three case distinctions, and provides tight small-n bounds and a direct computation for the notable case X^{[2]} when k=2. The results extend to nested Hilbert schemes X^{[n,n+1]} for large n, illustrating how pullbacks and diff-divisors control positivity. The work highlights how stability-wall analysis (Bayer–Macrì) and Mori-theoretic considerations yield precise birational-geometric information for Hilbert and nested Hilbert schemes on K3 surfaces, with potential generalizations to other surface classes.

Abstract

We illustrate the typical usage of Bayer and Macrì's Positivity Lemma to compute the nef cones of the Hilbert schemes by combining the Bridgeland stability methods (for large ) and classical methods (for small ). We use Mori dream K3 surfaces of Picard rank as our working example. We also compute the nef cones of the nested Hilbert schemes for such , for large .

Paper Structure

This paper contains 14 sections, 20 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. What is known
  3. What we do in this paper
  4. Preliminaries
  5. K3 surfaces
  6. Hilbert schemes of points
  7. Nested Hilbert schemes of points
  8. Bridgeland stability conditions
  9. Strategy:
  10. The nef cones
  11. The nef cone for large $n$
  12. Bounds for small $n$
  13. A direct computation of the nef cone of $X^{[2]}$
  14. Some closing remarks and possible future directions

Key Result

Theorem 1.1

The nef cone of the Hilbert scheme $X^{[n]}$ can be described as follows:

Theorems & Definitions (29)

  • Theorem 1.1: Theorem \ref{['thm:Nef_cone_case1']} and Theorem \ref{['thm:Nef_cone_case2and3']}
  • Proposition 1.2: Proposition \ref{['prop:bound_for_small_n_case1']}
  • Theorem 1.3: Theorem \ref{['thm:Nef_cone_nk2']}
  • Corollary 1.4
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • ...and 19 more