Line bundles on Contractions of $\overline{\rm{M}}_{0,n}$ via Coinvariant Divisors

Abstract

Using representations of vertex operator algebras, we describe the line bundles on a wide range of contractions of $\overline{\rm{M}}_{0,n}$, the moduli space of stable $n$-pointed rational curves, by proving a stronger version of the contraction theorem for these morphisms. These include the celebrated constructions of Kapranov, Keel, and Knudsen. Our main result suggests that while many so-called F-curves are not $K_X$-negative, they exhibit behavior similar to $K_X$-negative curves. This reveals for instance, a distinguished property of Knudsen's construction $f_{\text{Knu}}:\overline{\rm{M}}_{0,n}\to \overline{\rm{M}}_{0,n-1}\times_{\overline{\rm{M}}_{0,n-2}}\overline{\rm{M}}_{0,n-1}$, allowing for the classification of all possible factorizations of $f_{\text{Knu}}$, as well as further applications, and generalizations.