Line bundles on contractions of M̄0,n via coinvariant divisors

Using representations of affine Lie algebras, we describe line bundles on a broad class of contractions of M̄0,n M ¯ 0 , n , the moduli space of stable n -pointed rational curves, and show a variant of the cone and 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 KX K X -negative, they exhibit behavior similar to KX K X -negative curves. This reveals a distinguished property of Knudsen’s construction fKnu:M̄0,n→ M̄0,n-11.111pt× _M̄0,n-21.111ptM̄0,n-1 f Knu : M ¯ 0 , n → M ¯ 0 , n - 1 × M ¯ 0 , n - 2 M ¯ 0 , n - 1 , allowing for the classification of all possible factorizations of fKnu f Knu , as well as further applications.