#### Hall 123

#### Numerically effective divisors on the moduli space of curves from conformal field theory

#### Prakash Belkale

##### University of North Carolina

*Recent work of Fakhruddin has refocussed attention on conformal*

block divisors on moduli spaces of marked curves, in particular to

the birational geometry of moduli spaces of genus zero curves with marked

points.

Conformal blocks (which depend on a Lie group and n representations) give

an interesting family of numerically effective divisors (nef) on the

moduli

of $n$-pointed curves, and hence relate to well known conjectures on nef

cones of moduli spaces of curves. After reviewing moduli of curves and

conformal blocks, I will describe joint work with Angela Gibney and

Swarnava Mukhopadhyay where we study the higher level theory of these

divisors:

in particular producing vanishing theorems, new symmetries and

non-vanishing properties of these divisors (one of our tools is the

relation to quantum

cohomology of Grassmannians). These properties are then applied to the

study of moduli spaces.

Done