preprint submitted Loïs Faisant author 26ca6926-5797-11ee-9232-f8b51bd19631 TiBr department IST-BRIDGE: International postdoctoral program project 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. 2025 eng 2502.1170410.48550/ARXIV.2502.11704 L. Faisant, ArXiv (n.d.). Faisant L. Motivic counting of rational curves with tangency conditions via universal torsors. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/ARXIV.2502.11704">10.48550/ARXIV.2502.11704</a> L. Faisant, “Motivic counting of rational curves with tangency conditions via universal torsors,” <i>arXiv</i>. . Faisant, Loïs. “Motivic Counting of Rational Curves with Tangency Conditions via Universal Torsors.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/ARXIV.2502.11704">https://doi.org/10.48550/ARXIV.2502.11704</a>. Faisant L. Motivic counting of rational curves with tangency conditions via universal torsors. arXiv, 2502.11704. Faisant, Loïs. “Motivic Counting of Rational Curves with Tangency Conditions via Universal Torsors.” <i>ArXiv</i>, 2502.11704, doi:<a href="https://doi.org/10.48550/ARXIV.2502.11704">10.48550/ARXIV.2502.11704</a>. Faisant, L. (n.d.). Motivic counting of rational curves with tangency conditions via universal torsors. <i>arXiv</i>. <a href="https://doi.org/10.48550/ARXIV.2502.11704">https://doi.org/10.48550/ARXIV.2502.11704</a> 190552025-02-18T13:34:07Z2025-04-14T07:54:52Z