Motivic counting of rational curves with tangency conditions via universal torsors

Using the formalism of Cox rings and universal torsors, we prove a decomposition of the Grothendieck motive of the moduli space of morphisms from an arbitrary smooth projective curve to a Mori Dream Space (MDS). For the simplest cases of MDS, that of toric varieties, we use this decomposition to prove an instance of the motivic Batyrev–Manin–Peyre principle for curves satisfying tangency conditions with respect to the boundary divisors, often called Campana curves.