Motivic counting of rational curves with tangency conditions via universal torsors

TL;DR

This paper employs Cox rings and universal torsors to decompose the Grothendieck motive of moduli spaces of morphisms from smooth projective curves to Mori Dream Spaces, advancing understanding of rational curves with tangency conditions.

Contribution

It introduces a new motive decomposition framework for these moduli spaces, applying it specifically to toric varieties to verify the motivic Batyrev--Manin--Peyre principle for Campana curves.

Findings

Decomposition of the Grothendieck motive for moduli spaces of morphisms.

Verification of the motivic Batyrev--Manin--Peyre principle for toric varieties.

Application to counting rational curves with tangency conditions.

Abstract

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.