Boston Algebraic Geometry Day on Moduli of Curves

April 20, 2024 at Tufts University

organized by Dawei Chen, Aaron Landesman, Carl Lian, Montserrat Teixidor i Bigas

Speakers: Daebeom Choi, Hannah Larson, Miguel Moreira, Isabel Vogt

Location: Tufts University, Joyce Cummings Center, room 270

Directions: Here is the Tufts campus map.

Tufts is easily accessible by public transit: the Medford/Tufts green line station is right below the Joyce Cummings Center, and the Davis square red line station is a 20-25 minute walk away. Various bus lines serve the Tufts campus, and some street parking is available nearby.

We have limited funding to reimburse travel for junior participants. Please contact the organizers if you would like to attend but require financial support.

Schedule 

9:00-9:30 Check-in and refreshments
9:30-10:30 Isabel Vogt, Introduction to the moduli space of stable curves
10:30-11:00 Coffee break
11:00-12:00 Miguel Moreira, Cohomology and Chow rings of moduli spaces of curves
12:00-2:00 Lunch (not provided)
2:00-3:00 Daebeom Choi, Complete subvarieties of M_{g,n} and a lifting problem
3:00-3:30 Coffee break
3:30-4:30 Hannah Larson, The cohomology of M_{g,n}-bar and semi-tautological extensions

Abstracts

Isabel Vogt (Brown): Introduction to the moduli space of stable curves

In the study of smooth curves of genus g, it is often helpful or even essential to study limits of smooth curves that acquire singularities.  This talk will be an introduction to the compact moduli space of stable curves.  We will motivate the definition of stable curves and see where compactness comes from via an example-driven introduction to the stable reduction algorithm.

Miguel Moreira (MIT): Cohomology and Chow rings of moduli spaces of curves

This talk will be about the cohomology and Chow rings of moduli spaces of smooth/stable curves. The study of such rings goes back to Mumford in the early 80s and has been an active topic of research since then, with new and exciting results even today. I will introduce the tautological subring, which contains the most natural classes one can define on cohomology/Chow of moduli spaces of curves. I will try to give an account of what kind of questions concerning these rings have been asked, and what is known about them: are all classes tautological, what relations are there among tautological classes, how can we do interesction theory?

Daebeom Choi (Penn): Complete subvarieties of M_{g,n} and a lifting problem

Finding the maximal dimension of complete subvarieties of the moduli space of smooth n-pointed curves of genus g is a long-standing open problem. Here we show that for g ≥ 3*2^{d-1}, if the characteristic of the base field is greater than 2, then M_g contains a complete subvariety of dimension d. Furthermore, in positive characteristic, we construct a complete surface in M_{g,n} for g ≥ 3 and n ≥ 1, which contain a general point. These results follow from the proofs of the lifting conjectures, introduced here. In particular, we translate the existence of complete subvarieties to properties of line bundles on M_{g,n}. Our method reframes Zaal's approach, with increased efficiency via Keel's results on semi-ample line bundles in positive characteristic. This method demonstrates the difference in the geometry of moduli spaces between characteristic 0 and characteristic p. 

Hannah Larson (Berkeley): The cohomology of M_{g,n}-bar and semi-tautological extensions 

The cohomology ring of M_{g,n}-bar possesses a distinguished subring called the tautological ring (defined in Miguel's talk). In this talk, I'll ask the question: which cohomology groups H^k(M_{g,n}-bar) are generated by tautological classes? And when they are not, how can we better understand them? This leads to the new concept of semi-tautological extensions, which provide a framework for working with non-tautological classes in a controlled way. This is joint work with Samir Canning and Sam Payne.

Statement of Inclusion

We are dedicated to upholding the principles and practices of equal opportunity, affirmative action, and nondiscrimination. Our commitment extends to fostering an inclusive environment for all conference participants. Tufts University's related policies are applicable to all conference  participants, see here. If there are any concerns or requests for assistance related to discrimination or harassment, please reach out to one of the organizers.