Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866917364924153856 |
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| author | Faisant, Loïs |
| author_facet | Faisant, Loïs |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_11898 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups Faisant, Loïs Algebraic Geometry Number Theory 14H10, 14G10 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. |
| title | Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups |
| topic | Algebraic Geometry Number Theory 14H10, 14G10 |
| url | https://arxiv.org/abs/2106.11898 |