Chennai Mathematical Institute


11 am, Seminar Hall
Moduli Spaces of maps between genus-zero curves, with specified ramification

Rohini Ramadas
University of Michigan, Ann Arbor, USA.


How many degree four maps are there from P^1 to P^1, with one fully ramified point and three other simple ramification points, up to reparametrising source and target P^1s? If we fix the four-point branch locus in the target P^1, there are four such maps. In fact, the set W of all such maps forms a smooth variety that is a degree four cover of the moduli space M_{0,4} parametrizing ways to mark four distinct branch points on the target P^1. W has a compactification W(bar), which parametrises "admissible covers" between nodal genus-zero curves, and which admits a finite degree four map to M_{0,4}(bar), the moduli space of stable curves.

I will introduce admissible covers and their moduli via concrete examples. In the context of these examples I will discuss the maps

{admissible covers} -> M_{0,m}(bar) of marked target curves

{admissible covers} -> M_{0,n}(bar) of (the stabilization of) marked source curves

I will also talk about automorphisms of admissible covers and corresponding orbifold structure of their moduli spaces.

Moduli of admissible covers were first introduced by Harris and Mumford in order to define subvarieties of M_{g}(bar); related spaces have been used to find relations in the cohomology ring of M_{g,n} and M_{g,n}(bar). I am working on a project with Professors Sarah Koch and David Speyer.