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.