# Moduli spaces and blow-up constructions for stacks

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

Moduli spaces arise naturally in classification problems in algebraic geometry, and form a central ingredient in enumerative geometry (where we want to count objects of different types). A typical such problem, for example the classification of nonsingular projective curves up to isomorphism, can be resolved into two basic steps. The first step is to find as many discrete invariants as possible. The second step is to fix the values of all the discrete invariants and try to construct a moduli space; that is, an algebraic variety (or some more general geometric object) whose points correspond in a natural way to the equivalence classes of the objects to be classified. In enumerative geometry the aim is then to construct virtual fundamental classes on suitable moduli spaces and use them to evaluate cohomology classes.

What is meant by 'natural' here can be made precise given suitable notions of families of objects parametrised by base spaces and of equivalence of families. A fine moduli space is a base space for a universal family of the objects to be classified, but typically the existence of a fine moduli space is too much to hope for, unless we are willing to replace spaces by stacks. A moduli stack may have an associated coarse moduli space, satisfying slightly weaker conditions, but it is often the case that not even a coarse moduli space will exist. Typically, 'unstable' objects must be left out in order for a moduli space to exist.

The notion of stability here can depend on parameters which may be varied, and in enumerative geometry there are 'wall-crossing formulas' which describe the dependence on such parameters. Typically there is constancy on 'chambers' but things change in a prescribed way as walls between chambers are crossed.

In order to study moduli stacks, construct coarse moduli spaces and define and calculate enumerative invariants, an important tool is to have analogues for stacks of the blow-up construction for schemes which is used throughout algebraic geometry. A recent paper by Edidin and Rydh gives a blow-up construction which applies to an Artin stack with a stable good moduli space. Frances Kirwan is working with former students Victoria Hoskins and Joshua Jackson and current student Eloise Hamilton on a generalisation of the Edidin-Rydh construction which applies to much more general Artin stacks, while Dominic Joyce is working on a generalisation which allows the construction of the virtual fundamental classes needed in enumerative geometry. The aim of this research project is to extend these constructions and to exploit them, in particular in the case of surfaces.

This research is likely to have impact in the study of moduli stacks and wall-crossing formulas in enumerative geometry. It is not directly related to any of the EPSRC's priority research areas. The research methodology using blow-up constructions for stacks is very new, although based on well established constructions for algebraic varieties and schemes. No companies or collaborators are involved.

This project falls within the EPSRC Geometry and Topology research area

What is meant by 'natural' here can be made precise given suitable notions of families of objects parametrised by base spaces and of equivalence of families. A fine moduli space is a base space for a universal family of the objects to be classified, but typically the existence of a fine moduli space is too much to hope for, unless we are willing to replace spaces by stacks. A moduli stack may have an associated coarse moduli space, satisfying slightly weaker conditions, but it is often the case that not even a coarse moduli space will exist. Typically, 'unstable' objects must be left out in order for a moduli space to exist.

The notion of stability here can depend on parameters which may be varied, and in enumerative geometry there are 'wall-crossing formulas' which describe the dependence on such parameters. Typically there is constancy on 'chambers' but things change in a prescribed way as walls between chambers are crossed.

In order to study moduli stacks, construct coarse moduli spaces and define and calculate enumerative invariants, an important tool is to have analogues for stacks of the blow-up construction for schemes which is used throughout algebraic geometry. A recent paper by Edidin and Rydh gives a blow-up construction which applies to an Artin stack with a stable good moduli space. Frances Kirwan is working with former students Victoria Hoskins and Joshua Jackson and current student Eloise Hamilton on a generalisation of the Edidin-Rydh construction which applies to much more general Artin stacks, while Dominic Joyce is working on a generalisation which allows the construction of the virtual fundamental classes needed in enumerative geometry. The aim of this research project is to extend these constructions and to exploit them, in particular in the case of surfaces.

This research is likely to have impact in the study of moduli stacks and wall-crossing formulas in enumerative geometry. It is not directly related to any of the EPSRC's priority research areas. The research methodology using blow-up constructions for stacks is very new, although based on well established constructions for algebraic varieties and schemes. No companies or collaborators are involved.

This project falls within the EPSRC Geometry and Topology research area

## People |
## ORCID iD |

Dominic Joyce (Primary Supervisor) | |

Alexei Latyntsev (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/R513295/1 | 01/10/2018 | 30/09/2023 | |||

2099935 | Studentship | EP/R513295/1 | 01/10/2018 | 31/03/2022 | Alexei Latyntsev |