Moduli of del Pezzo Surfaces Seminar

Welcome to the homepage for the moduli of del Pezzo surfaces seminar! I will try to keep this updated, but please also check the MS Digest for updates.

Time and location:

Wednesdays 12pm-1pm, in the Greg Hjorth Seminar Room (Peter Hall 107).
Exceptions: Moved to be 12:30pm-1:30pm 17th September.

In this seminar, we will read the paper “Artin Groups and the Fundamental Groups of Some Moduli Spaces” by Eduard Looijenga. In the paper, groups related to Coxeter graphs appear as fundamental groups of moduli spaces. We will develop an understanding of these groups, find a toric space for which the groups appear as fundamental groups, and then show that the toric spaces embed into the moduli spaces we want to understand, the embedding giving an isomorphism of orbifold fundamental groups.

The goal is to get experience with moduli spaces and the algebraic geometry of surfaces, so there will be room in the schedule to take a closer look at related topics of interest, for example the famous 27 lines on a cubic surface.

Newcomers to algebraic geometry are welcome!

Resources:

Looijenga - Artin Groups and the Fundamental Groups of Some Moduli Spaces: https://webspace.science.uu.nl/~looij101/affartinml2a.pdf Beauville - Complex Algebraic Surfaces (Unimelb has institutional access): https://doi.org/10.1017/CBO9780511623936

Schedule:

• (6.08.25) Corey Lionis: Introduction and Motivation

  I will give an overview of the goals of the paper and define the objects we will study: moduli spaces and their orbifold fundamental groups. We will talk about a particular class of examples that lead to the moduli of del Pezzo surfaces. I'll mention some of the ideas it would be helpful to expand on during the seminar, and if time permits I will run through the precise statements of the main results and the topological interpretation.

• (13.08.25) Corey Lionis:

  Since Alice is sick, I will review the hypersurface moduli space and explain some other results on fundamental groups of moduli spaces. I will also say something about how the fundamental group is known topologically.

• (20.08.25) Alice Rossiter: Artin Groups and Orbit Space

  Alice will discuss the different groups we will need to know and their structure. She might talk a bit about Artin braid groups to motivate the construction.

• (27.08.25) George Henderson-Walsh: Toric structure

  George will explain how we see the groups as fundamental groups.

• (17.09.25) Phoenix Pham: (Toric) blowups

  Phoenix will go through some examples of blowups and then specialise to the toric blowup we need to understand.

• (24.09.25) Dom Vlachos: Kodaira classifications and del Pezzo surfaces

  In this talk, Dom plans to discuss the definition of del Pezzo surface and explain which surfaces are del Pezzos of given degree. He will also discuss the Kodaira classification of surfaces and of singular fibres, explaining where del Pezzos fit in.

• (08.10.25) Adam Monteleone: The moduli space of marked del Pezzo surfaces

• Adam will go over the definition and some of the properties of this moduli space.

(??.??.??) ??: Toric structure and toric extension (Will go back to this in future if time permits)

• We'll discuss a toric variety interpretation interpretation of the spaces George introduced us to, and then look at how the orbifold fundamental group behaves under extending from an open subscheme to a complementary Weil divisor.
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