Irreducibility and degree of the logarithmic discriminant of M_{0,n}

Show that the logarithmic discriminant of the moduli space M_{0,n} is an irreducible hypersurface and determine its degree, as conjectured by Kayser–Kretschmer–Telen (Conjecture 1).

Background

The logarithmic discriminant controls collisions of solutions to logarithmic critical point equations on M_{0,n}, directly relevant to CHY and string limits. Its geometric properties (irreducibility, degree) encode deep combinatorial and moduli-space information. Establishing irreducibility and computing the degree would advance understanding of scattering equation degenerations and associated arrangement complements.