Using representations of affine Lie algebras, we describe line bundles on a broad class of contractions of \(\overline{\textrm{M}}_{0,n}\) , 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 \(K_X\) -negative, they exhibit behavior similar to \(K_X\) -negative curves. This reveals a distinguished property of Knudsen’s construction \(f_{\textrm{Knu}}:\overline{\textrm{M}}_{0,n}\rightarrow \overline{\textrm{M}}_{0,n-1}\hspace{1.111pt}{\times }_{\overline{\textrm{M}}_{0,n-2}}\hspace{1.111pt}\overline{\textrm{M}}_{0,n-1}\) , allowing for the classification of all possible factorizations of \(f_{\textrm{Knu}}\) , as well as further applications.
