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Nef Cones of the Hilbert Schemes of Points on Generalized Cayley K3 Surfaces

Chiwon Yoon

TL;DR

The paper analyzes nef and Mori cones for Hilbert schemes of points on a Cayley K3 surface and its generalizations $S_a$ by blending lattice theory via the BBF form, automorphism actions, and Bridgeland stability. It provides an explicit nef-cone description for the Hilbert square $S_1^{[2]}$, including a rational polyhedral fundamental domain for $\mathrm{Aut}(S_1^{[2]})$, and extends to large $n$ by describing the nef cone of $S_a^{[n]}$ through walls determined by contracted curves and extremal divisors $D_{k,t}$. The results reveal a concrete wall-and-chamber structure governed by automorphisms and stability data, with irrational limiting Mori rays and explicit criteria on $n$ that ensure complete cone descriptions. Overall, the work demonstrates how lattice theory, automorphism groups, and Bridgeland stability interact to give detailed geometric descriptions for a non-Mori-dream family of hyperkähler manifolds.

Abstract

We study the nef cones and fundamental domains of Hilbert schemes of points on the Cayley K3 surface $S$ and its generalizations $S_a$. For the Hilbert square $S^{[2]}$, we explicitly compute the nef cone and describe a fundamental domain using the automorphisms of $S^{[2]}$ and lattice-theoretic methods. For higher Hilbert schemes $S_a^{[n]}$, we determine the nef cones using Bridgeland stability methods that identify the contracted curves defining walls and the divisors generating the extremal rays.

Nef Cones of the Hilbert Schemes of Points on Generalized Cayley K3 Surfaces

TL;DR

The paper analyzes nef and Mori cones for Hilbert schemes of points on a Cayley K3 surface and its generalizations by blending lattice theory via the BBF form, automorphism actions, and Bridgeland stability. It provides an explicit nef-cone description for the Hilbert square , including a rational polyhedral fundamental domain for , and extends to large by describing the nef cone of through walls determined by contracted curves and extremal divisors . The results reveal a concrete wall-and-chamber structure governed by automorphisms and stability data, with irrational limiting Mori rays and explicit criteria on that ensure complete cone descriptions. Overall, the work demonstrates how lattice theory, automorphism groups, and Bridgeland stability interact to give detailed geometric descriptions for a non-Mori-dream family of hyperkähler manifolds.

Abstract

We study the nef cones and fundamental domains of Hilbert schemes of points on the Cayley K3 surface and its generalizations . For the Hilbert square , we explicitly compute the nef cone and describe a fundamental domain using the automorphisms of and lattice-theoretic methods. For higher Hilbert schemes , we determine the nef cones using Bridgeland stability methods that identify the contracted curves defining walls and the divisors generating the extremal rays.
Paper Structure (11 sections, 12 theorems, 92 equations, 2 figures)

This paper contains 11 sections, 12 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. K3 surfaces and their Hilbert schemes
  4. Cones of hyperkähler manifolds
  5. Cone conjecture
  6. Beauville involution
  7. Bridgeland stability conditions
  8. Generalized Fibonacci sequences
  9. The nef cone of $S_1^{[2]}$
  10. Fundamental domain
  11. Extremal rays of the nef cone for large $n$

Key Result

Theorem 1.1

(=Theorem thm:nef_cone_of_S^[2]) Every wall of the nef cone of the Hilbert square of the Cayley K3 surface $S_1$ is given by $(\iota\cdot e)^\perp$, where $\iota \in \mathop{\mathrm{Aut}}\nolimits(S^{[2]})$.

Figures (2)

  • Figure 4.1: The slice of the nef cone of $S_1^{[2]}$ where the coefficient of $h_1$ is fixed to be $1$
  • Figure 5.1: The slice of the nef cone of $S_1^{[6]}$ where the coefficient of $h_1$ is fixed to be $1$

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 2.1: Beauville--Bogomolov--Fujiki form Bea83bFuj87
  • Theorem 2.2: HT09
  • Conjecture 2.3: Morrison--Kawamata
  • Definition 2.4: Beauville involution
  • Remark 2.5
  • Theorem 2.6: Positivity Lemma BM14a
  • Theorem 2.7: Walls in the $(H,D)$-slice BHL+16
  • Proposition 2.8: Gieseker wall BHL+16
  • Proposition 2.9: BHL+16
  • ...and 13 more