Geometric Batyrev–Manin–Peyre for equivariant compactifications of additive groups

Building on previous works by Bilu, Chambert-Loir and Loeser, we study the asymptotic behaviour of the moduli space of sections of a given family over a smooth projective curve, assuming that the generic fiber is an equivariant compactification of a finite dimensional vector space. Working in a suitable Grothendieck ring of varieties, we show that the class of these moduli spaces converges, modulo an adequate normalisation, to a non-zero effective element, when the class of the sections goes arbitrary far from the boundary of the dual of the effective cone. The limit can be interpreted as a motivic Euler product in the sense of Bilu’s thesis. This result provides a positive answer to a motivic version of the Batyrev–Manin–Peyre conjectures in this particular setting.