This lecture course will introduce the moduli space of stable curves, with a focus on its intersection theory. This space is a compactification of the moduli space of smooth, projective curves, constructed by Deligne and Mumford. It has played an important role in algebraic geometry and has connections to many different parts of mathematics (e.g. graph theory, the theory of integrable systems and enumerative geometry).

In the course, we will recall basic properties of algebraic curves and define their moduli space. We will show how modern tools of algebraic geometry, such as deformation theory, can be used to study properties of this space (introducing those tools as we need them). The second part of the course will focus on the intersection theory in the singular cohomology rings of the moduli space of stable curves. These rings contain the so-called tautological rings, subrings with explicit generators indexed by decorated graphs, with explicit combinatorial rules for computing intersection products or relations between the generators.

Algebraic Geometry (roughly Hartshorne, Chapters 1-3, or as taught in Algebraic Geometry I and II), Singular Cohomology

- E. Arbarello, M. Cornalba, P. Griffiths: Geometry of Algebraic Curves, Volume II (Chapters X - XX)
- J. Kock, I. Vainsencher: An Invitation to Quantum Cohomology (Chapter 1)
- D. Zvonkine: An introduction to moduli spaces of curves and its intersection theory (Lecture notes)
- J. Harris, I. Morrison: Moduli of Curves
- P. Deligne, D. Mumford: The irreducibility of the space of curves of given genus
- E. Sernesi: Algebraic Curves and their Moduli (in: Moduli of curves, Springer 2016)

Instead of chopping the script up according to the lectures, here is the full Script in one file (version: 30/09/2020).

Lecture 1 (21st April) : Organizational remarks , Introduction and Motivation (Board)

Lecture 2 (28th April) : Moduli functors and fine moduli spaces (Board)

Lecture 3 (5th May) : Coarse moduli spaces, smooth and nodal curves (Board)

Lecture 4 (12th May) : Automorphisms of curves, families of curves, the moduli space of curves (Board)

Lecture 5 (19th May) : Properties of the moduli space of curves, Smooth curves in genus 0, Stable graphs (Board)

Lecture 6 (26th May) : Stratification of the moduli space of stable curves according to stable graph (Board)

Lecture 7 (2nd June) : Stable curves in genus 0, the forgetful morphism and the universal curve (Board)

Lecture 8 (9th June) : Stable curves in genus 1, Moduli stacks of curves (Board)

Lecture 9 (16th June) : More on stacks, Intersections of boundary strata in the moduli of curves (Board)

Lecture 10 (23rd June) : Morphisms of stable graphs, fibre products of gluing morphisms; Information about the exam (Board)

Lecture 11 (30th June) : Crash course in intersection theory (Board)

Lecture 12 (7th July) : Chern classes and excess intersection formulas, the tautological ring (Board)

Lecture 13 (14th July) : Decorated strata classes, products in the tautological ring; a panorama of known results (Board)

Lecture 14 (11th August) : Construction of the moduli space of curves, deformation theory (Board)

Lecture 15 (18th August) : Higher order deformations, local structure of the boundary (Board)

Due to the current situation, the lectures will only start on April 21 and will be held in an online format. More information on the exact format will follow.

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