Lecturer

Prof. Dr. Rahul Pandharipande

Coordinator

Younghan Bae

Exam

The example will happen on February 3rd. Here is the List of questions for the exam.

Lecture schedule

Due to the current situation, the lectures will be delivered entirely online via live Zoom sessions. Lectures will take place on

• Tuesday, 10:00-12:00
• Thursday, 13:15-14:00

Video recordings of the sessions will be provided.

Content

In this course we will study modern aspect of Abel-Jacobi theory.
Possible topics include:

• Basic knowledge of algebraic curves including classical Abel-Jacobi theory
• Moduli space of stable curves
• Moduli space of (relative) stable maps and the definition of double ramification cycles
• Tautological classes and Pixton's formula for double ramification cycles
• Proof of Pixton's formula
• Universal Abel-Jacobi theory over the universal Picard stacks

Here is the link to the webpage in Prof. Pandharipande's website.

Prerequisites

It is useful to have a basic knowledge of algebraic geometry.

Lecture Notes

Tuesdays:

• Classical Abel-Jacobi theory
• Moduli of curves
• A User guide to Sage program by Johannes Schmitt
• Moduli space of stable maps
• Proof of Pixton's formula

Thursdays:

• Cech cohomology, line bundles
• Divisors on curves, Abel-Jacobi theorem for elliptic curves
• Line bundles on a complex torus
• Projectivity of Jacobians
• Tautological classes on the moduli space of stable curves
• Intersecting boundary strata
• String and dilaton equations
• Line bundles and the dualizing sheaf on a nodal curve
• Introduction to the moduli space of stable maps
• Genus zero stable maps to a projective space
• Relative/rubber stable maps
• Pixton's double ramification cycle formula
• Virtual localization formula
• Moduli space of r th roots and Chiodo's formula

Previous Lectures

• ICM Plenary Lecture in Rio (2018) on Geometry of the moduli space of curves
• Shafarevich Seminar in Moscow (2020, Zoom) on Abel-Jacobi maps and double ramification cycles

Literature

Some books on basic theory of algebraic curves and Jacobians

• "Principles of algebraic geometry" by P. Griffith and J. Harris
• "Curves and their Jacobians" by D. Mumford
• "Moduli of curves" by J. Harris and I. Morrison

Some articles on the moduli space of stable curves, tautological classes

• A calculus for the moduli space of curves R. Pandharipande
• Geometry of the moduli of curves R. Pandharipande
• The irreducibility of the space of curves of given genus P. Deligne and D. Mumford
• Towards an Enumerative Geometry of the Moduli Space of Curves D. Mumford
• Course webpage on the moduli space of curves J. Schmitt

Some articles on the moduli space of stable maps

• Notes on stable maps and quantum cohomology by W. Fulton and R. Pandharipande
• Localization of virtual classes by T. Graber and R. Pandharipande
• "An invitation to quantum cohomology. Kontsevich's formula for rational plane curves" by J. Kock and I. Vainsencher

Some articles on the double ramification cycles and Pixton's formula

• Double ramification cycles on the moduli spaces of curves by F. Janda, R. Pandharipande, A. Pixton and D. Zvonkine
• Double ramification cycles with target varieties by F. Janda, R. Pandharipande, A. Pixton and D. Zvonkine

Some articles/notes on universal double ramification cycles

• Pixton’s formula and Abel-Jacobi theory on the Picard stack by Y. Bae, D. Holmes, R. Pandharipande, J. Schmitt and R. Schwarz
• Abel-Jacobi maps and Double ramification cycles by R. Pandharipande