250137 VO Moduli of n points on the projective line (2021W)

International Lecture Course Fall 2021:

"Moduli of n points on the projective line"

Outset: This course is designed for both students and (non-expert) researchers who wish to see how apparently remote mathematical theories come together and interlace. It will be given in digital form via Zoom from October 5th, 2021, until January 25, 2022.

Up to technical obstructions we plan to give the course on a transparent blackboard, the so-called lightboard, which allows the audience to simultaneously follow the hand-writing and the lecturer from the front.

The trailer for this course can be found on

https://ucloud.univie.ac.at/index.php/s/7kJ9XemV0kzA3Yq

The prospective schedule is Tuesdays, from 5 pm till 6.30. The link to the online transmission will be distributed in due time. If you are potentially interested, please send an e-mail to

herwig.hauser@univie.ac.at

or go to the course website

www.xxyyzz.cc

in order to receive updated information (you may unsubscribe at any moment).

Topic: The problem of constructing normal forms and moduli spaces for various geometric objects goes back (at least, and among many others) to the Italian geometers (Enriques, Chisini, Severi, ...). A highlight was reached in the 1960 and 70ies when Deligne, Mumford and Knudsen constructed the moduli space of stable curves of genus g. These spectacular works had a huge impact, though the techniques from algebraic geometry they applied were quite challenging.

In the course, we wish to offer a gentle and hopefully fascinating introduction to these results, restricting always to curves of genus
zero.

Contents: We start by discussing the concept of (coarse and fine) moduli spaces and universal families, providing also the philosophical background thereof: why is it natural to study such questions, and why the given axiomatic framework is the correct one? Once we have become familiar with these foundations (seeing many examples on the way), we will concentrate on n points in P1 and the action of PGL2 on them by Möbius transformations. This is part of classical projective geometry and very beautiful. As long as the n points are pairwise distinct, things are easy, and a moduli space is easily constructed. Things become tricky as the points start to move and thus become closer to each other until they collide and coalesce. What are the limiting configurations of the points one has to expect in this variation? This question has a long history –Grothendieck proposed a convincing answer: n-pointed stable curves. Then, Deligne, Mumford and Knudsen built a fascinating and multi-faceted theory for them, their famous compactification of M–(0,n).

We will take at the beginning a different approach by proposing an alternative version of limit. Namely, we embed the space of (PGL2-orbits of) n distinct points into a large projective space and then take limits therein via the Zariski-closure. That’s a one page construction of a compact space Xn. It opens the door to the theory of phylogenetic trees: they are certain finite graphs with leaves and inner vertices as a tree in a forest. Their combinatorial structure will become the guiding principle to design many proofs for our moduli spaces. Working with phylogenetic trees can be a very pleasing occupation: we will draw, glue, cut and compose these trees and thus get surprising constructions and insights.

At that point a miracle happens: When considering the above Zariski-closure Xn and the associated phylogenetic trees, the stable curves of Grothendieck, Deligne, Mumford, Knudsen pop up on their own. We don’t even have to define them – they are just
there. So the circle closes up, as we then get an isomorphism from Xn to the compactification of M-(0,n) . In this way our journey is now able to reprove many of the classical results in an easy going and appealing manner.