Line bundles on Contractions of M̄0,n via Conformal Block Divisors

The moduli space of stable curves of genus g with n marked points, M̄g,n, is a central object in algebraic geometry, and plays a crucial role in $2$-dimensional conformal field theory. In this paper, we apply the sheaf of coinvariants and conformal block divisors to study the geometry of M̄0,n. The main theorem characterizes the line bundles on certain contractions of M̄0,n via F-curves, using Fakhruddin's basis for the Picard group given by conformal block divisors. This reveals a distinguished property of Knudsen's construction of M̄0,n. As a notable consequence of this property, using the global generation of sheaves of coinvariants by Fakhruddin, we refine Knudsen's construction by describing every possible contraction of M̄0,n over it. An application of this refinement is given, and its potential for generalization is discussed.