Geometry Seminar

 

University of Tor Vergata, Department of Mathematics

27^th  of February 2024, 14:30-16:00, room “Dal Passo”

 

 

 

 

 

 

 

Equivariant localization in the absence of a group action

 

Sam Molcho
ETH

 

 

 

 

 

 

Consider the moduli space of stable, n-marked curves M and the tautological
subring R^*(M) of its Chow ring. The standard calculus for R^*(M) is based
on the ``strata algebra" SA(M), which is constructed via the inductive
structure of the boundary of M and the excess intersection formula, and in
which calculations are expressed in terms of ``graph sums". In this talk I
will discuss a new calculus for R^*(M), based on the introduction of a new
ring L^*(M), built out of tropical geometry, and in which several standard
calculations simplify significantly. I will explain how the comparison
between SA and L is analogous to the comparison between equivariant
cohomology and equivariant cohomology of the fixed locus in GKM theory.
Finally, I will sketch how this idea can be used to give explicit formulas
for the Brill-Noether cycles -- informally, the cycles on M parametrizing
curves on which a line bundle of the form \omega^k(sum a_ix_i) has at least
r+1 linearly independent sections. This is a joint work with M. Abreu and
N. Pagani.

 

 

 

 

 

 

 

 

 

 

 

 

This talk is part of the activity of the MIUR Excellence Department Projects MathMod@TOV, and the PRIN 2022 Moduli Spaces and Birational Geometry