Expansions for Hilbert Schemes: Background on Tropical Perspective

11 Jun 2024



2. Background on tropical perspective

We briefly introduce here the language of tropical and logarithmic geometry in the context of this problem. For more details on the contents of this section, see the article [Log], lecture notes [Ran22a], as well as the first section of [MR20].

2.1 Tropicalisation and expansion

Subdivisions of the tropicalisation define expansions of X. In the following, we will want to study possible birational modifications of the scheme X around the divisor D. In the tropical language, these are expressed as subdivisions.

A subdivision of the tropicalisation defines a birational modification of X in the following way. The subdivision

2.2 Maulik-Ranganathan construction

We will briefly recall some key points of [MR20]. The aim of their work is to study the moduli space of ideal sheaves of fixed numerical type which meet the boundary divisor transversely. Some key motivations for the study of such an object come from enumerative geometry. For example, a common method used to address problems of curve counting in a given smooth variety is to degenerate this variety to a singular union of simpler irreducible components. The property of transversality is then crucial to ensure that all interesting behaviour of the ideal sheaves on the degenerate object occurs with support in the interior of the simpler irreducible components, which allows us to study it with more ease. One of the main difficulties with this approach is that often, as in this setting, the space of transverse ideal sheaves with respect to D is non-compact. Constructing the appropriate compactification will yield a space which is flat and proper over C. In [MR20], Maulik and Ranganathan formulate the Donaldson-Thomas theory of the pair (X, D), starting by constructing compactifications of the space of ideal sheaves in X transverse to D.

We discuss [MR20] specifically with respect to the case which interests us here, namely that of a degeneration X ! C as described above, where we seek to study the moduli space of ideal sheaves with fixed constant Hilbert polynomial m, for some m ∈ N with respect to the boundary divisor D = X0. The key idea is to construct the tropicalisation of X, denoted ΣX , and a corresponding tropicalisation map, which is used to understand how to obtain the desired transversality properties in our compactifications.

Existence and uniqueness of transverse limits. Maulik and Ranganathan introduce notions of dimensional transversality and strong transversality, which, in the specific case of Hilbert schemes of points, happen to be equivalent to Li-Wu stability (see Section 5.3 for a definition of this stability condition). In general for higher dimensional subschemes this will not be the case, however.

This operation results in non-uniqueness, as we are making a choice of polyhedral subdivision and there is in general no canonical choice.

The addition of these tube vertices in the tropicalisation means that there are more potential components in each expansion, which interferes with the previously set up uniqueness results. Indeed, recall that trop(Z ◦ ) gave us exactly the right number of vertices in the dual complex in order for each family of subschemes Z ◦ ⊂ X◦ to have a unique limit representative. Therefore, to reflect this, Donaldson-Thomas stability asks for subschemes to be DT stable if and only if they are tube schemes precisely along the tube components. We say that a 1-dimensional subscheme is a tube if it is the schematic preimage of a zerodimensional subscheme in D. In the case of Hilbert schemes of points, this condition will translate simply to a 0-dimensional subscheme Z being DT stable if and only if no tube component contains a point of the support of Z and every other irreducible component expanded out by our blow-ups contains at least one point of the support of Z.

Maulik and Ranganathan define a subscheme to be stable if it is strongly transverse

and DT stable. For fixed numerical invariants the substack of stable subschemes in the space of expansions forms a Deligne-Mumford, proper, separated stack of finite type over C.

Comparison with the results of this paper. The construction we present in this paper has the surprising property that we do not need to label any components as tubes in order for the stack of stable objects we define to be proper. This is an artifact of the specific choices of blow-ups to be included in our expanded degenerations. The work of Maulik and Ranganathan shows us that this is not expected in general. As mentioned in Section 1.3, we will discuss in an upcoming paper how to construct proper stacks of stable objects in cases where different choices of expansions are made and it becomes necessary for us as well to introduce a Donaldson-Thomas stability condition.

This paper is available on arxiv under CC 4.0 license.