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Content
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 complex 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.
We will mainly follow lecture notes of a course by Johannes Schmitt: https://johannesschmitt.gitlab.io/ModCurves/Script.pdf.
The other main sources are the books:
E. Arbarello, M. Cornalba, P. Griffiths: "Geometry of Algebraic Curves, Volume II".
J. Harris, I. Morrison: "Moduli of Curves".
Prerequisites
Algebraic Geometry (roughly Hartshorne, Chapters 1-3), Singular Cohomology |
