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Good models of Hilbert schemes of points over semistable degenerations

Calla Tschanz

Abstract

In this paper, we explore different possible choices of expanded degenerations and define appropriate stability conditions in order to construct good degenerations of Hilbert schemes of points over semistable degenerations of surfaces, given as proper Deligne-Mumford stacks. These stacks provide explicit examples of constructions arising from the work of Maulik and Ranganathan. This paper builds upon and generalises previous work in which we constructed a special example of such a stack. We also explain how these methods apply to constructing minimal models of type III degenerations of hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces.

Good models of Hilbert schemes of points over semistable degenerations

Abstract

In this paper, we explore different possible choices of expanded degenerations and define appropriate stability conditions in order to construct good degenerations of Hilbert schemes of points over semistable degenerations of surfaces, given as proper Deligne-Mumford stacks. These stacks provide explicit examples of constructions arising from the work of Maulik and Ranganathan. This paper builds upon and generalises previous work in which we constructed a special example of such a stack. We also explain how these methods apply to constructing minimal models of type III degenerations of hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces.
Paper Structure (20 sections, 29 theorems, 58 equations, 7 figures)

This paper contains 20 sections, 29 theorems, 58 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The previous construction
  4. Tropical perspective
  5. The expanded constructions
  6. Scheme construction
  7. The group action and linearisation
  8. The stack construction
  9. The stack of expansions
  10. The family $\mathfrak{X}'$ over $\mathfrak{C}'$.
  11. Stability conditions
  12. Universal closure
  13. Constructing separated stacks.
  14. Non-algebraic proper stacks
  15. Choices of proper algebraic stacks.
  16. ...and 5 more sections

Key Result

Theorem 1.1

The stacks $\overline{\mathfrak{N}^m_{\mathop{\mathrm{LW}}\nolimits}}$ and $\overline{\mathfrak{N}^m_{\mathop{\mathrm{SWS}}\nolimits}}$ are proper over $C$ and have finite automorphisms.

Figures (7)

  • Figure 1: Geometric and tropical pictures of the special fibre $X_0$.
  • Figure 2: Geometric and tropical picture at $t_i=t_j=0$ for $i,j\in A\setminus B$ in $X[A,B]$.
  • Figure 3: Geometric and tropical picture at $t_1 = t_2 = t_3 =0$ in $X[\{1,2\},\{3\}]$.
  • Figure 4: Three different stable associated pairs for the same tropical picture.
  • Figure 5: The picture on the right is not a limit of the one on the left.
  • ...and 2 more figures

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Proposition 3.2
  • ...and 69 more