Expansions for Hilbert Schemes: Abstract and Intro

11 Jun 2024




The aim of this paper is to extend the expanded degeneration construction of Li and Wu to obtain good degenerations of Hilbert schemes of points on semistable families of surfaces, as well as to discuss alternative stability conditions and parallels to the GIT construction of Gulbrandsen, Halle and Hulek and logarithmic Hilbert scheme constructions of Maulik and Ranganathan. We construct a good degeneration of Hilbert schemes of points as a proper Deligne-Mumford stack and show that it provides a geometrically meaningful example of a construction arising from the work of Maulik and Ranganathan.

1. Introduction

The study of moduli spaces is a central topic in algebraic geometry; among moduli spaces, Hilbert schemes form an important class of examples. They have been widely studied in geometric representation theory, enumerative and combinatorial geometry and as the two main examples of hyperk¨ahler manifolds, namely Hilbert schemes of points on K3 surfaces and generalised Kummer varieties. A prominent direction in this area is to understand the local moduli space of such objects and, in particular, the ways in which a degeneration of smooth Hilbert schemes may be given a modular compactification.

For example, we may consider the geometry of relative Hilbert schemes on a degeneration whose central fibre has normal crossing singularities. We may then ask how the singularities of such a Hilbert scheme may be resolved while preserving certain of its properties or how it may be expressed as a good moduli space. This then becomes a compactification problem with respect to the boundary given by the singular locus. Historically, an important method used in moduli and compactification problems has been Geometric Invariant Theory (GIT). More recently, the work of Maulik and Ranganthan [MR20] has explored how methods of tropical and logarithmic geometry can be used to address such questions for Hilbert schemes. This builds upon previous work of Li [Li13] and Li and Wu [LW15] on expanded degenerations for Quot schemes and work of Ranganathan [Ran22b] on logarithmic Gromov-Witten theory with expansions.

Briefly stated, the aim of this paper is to provide explicit examples of such compactifications and explore the connections between these methods.

1.1 Basic setup

As is mentioned in Section 1.3, this type of construction can be applied to construct type III degenerations of Hilbert schemes of points on K3 surfaces. This will be described in future work.

1.2 Previous work in this area

Following on from [LW15], Gulbrandsen, Halle and Hulek [GHH19] present a GIT version of the above construction in the case of Hilbert schemes of points. They construct an explicit expanded degeneration, i.e. a modified family over a larger base, whose fibres correspond to blow-ups of components of X0 in the family. They present a linearised line bundle on this space for the natural torus action and they are able to show that in this case the Hilbert-Mumford criterion simplifies down to a purely combinatorial criterion. Using this, they impose a GIT stability condition which recovers the transverse zero-dimensional subschemes of Li and Wu and prove that the corresponding stack quotient is isomorphic to that of Li and Wu. A motivation for this work was to construct type II degenerations of Hilbert schemes of points on K3 surfaces. Indeed, type II good degenerations of K3 surfaces present these types of singularities in the special fibre, which is a chain of surfaces intersecting along smooth curves.

There is more recent work of Maulik and Ranganathan [MR20], building upon earlier ideas of Ranganathan [Ran22b] and results of Tevelev [Tev07], in which they use techniques of logarithmic and tropical geometry to construct appropriate expansions of X ! C. This allows them to define moduli stacks of transverse subschemes starting from the case where X0 is any simple normal crossing variety. They show that the stacks thus constructed are proper and Deligne-Mumford. For more details on this, see Section 2.2.

1.3 Main results

Let X ! C be a semistable degeneration of surfaces. In the following sections, we propose explicit constructions of expanded degenerations and stacks of stable length m zero-dimensional subschemes on these expanded families, which we show to have good properties.

Allowing for different choices of expansions. In this paper, we discuss only a specific choice of model for the Hilbert scheme of points which we call the canonical moduli stack. In upcoming work, we will investigate how these methods can be extended to describe other choices of models. We will consider an approach which parallels work of Kennedy-Hunt on logarithmic Quot schemes [Ken23], as well as recover certain geometrically meaningful choices of moduli stacks arising from the methods of Maulik and Ranganathan [MR20]. In particular, we will discuss how tube components and DonaldsonThomas stability enter the picture in these more general cases (see Section 2.2 for definitions).

1.4 Organisation

We start, in Section 2, by giving some background on logarithmic and tropical geometry, and an overview of the work of Maulik and Ranganathan from [MR20] which we will want to refer to in later sections. Then, in Section 3, we set out an expanded construction on schemes and, in 4, we discuss how various GIT stability conditions can be defined on this construction. In Section 5, we describe a corresponding stack of expansions and family over it, building on the expanded degenerations we constructed as schemes. In Section 6, we extend our stability conditions to this setting. We then show that the stacks of stable objects defined have the desired Deligne-Mumford and properness properties.

Acknowledgements. I would like to thank Gregory Sankaran for all his support throughout this project. Thank you also to my PhD examiners, Alastair Craw and Dhruv Ranganathan, for their many helpful comments. This work was undertaken while funded by the University of Bath Research Studentship Award. I am also grateful to Patrick Kennedy-Hunt and Thibault Poiret for many interesting conversations.

This paper is available on arxiv under CC 4.0 DEED license.