Conformal blocks and rational normal curves

Abstract

We prove that the Chow quotient parameterizing configurations of n n points in P d \mathbb {P}^d which generically lie on a rational normal curve is isomorphic to M ¯ 0 , n \overline {\mathcal {M}}_{0,n} , generalizing the well-known d = 1 d=1 result of Kapranov. In particular, M ¯ 0 , n \overline {\mathcal {M}}_{0,n} admits birational morphisms to all the corresponding geometric invariant theory (GIT) quotients. For symmetric linearizations, the polarization on each GIT quotient pulls back to a divisor that spans the same extremal ray in the symmetric nef cone of M ¯ 0 , n \overline {\mathcal {M}}_{0,n} as a conformal blocks line bundle. A symmetry in conformal blocks implies a duality of point-configurations that comes from Gale duality and generalizes a result of Goppa in algebraic coding theory. In a suitable sense, M ¯ 0 , 2 m \overline {\mathcal {M}}_{0,2m} is fixed pointwise by the Gale transform when d = m − 1 d=m-1 so stable curves correspond to self-associated configurations.