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Asymptotic behaviour of rational curves

By Loïs Faisant

Appears in collection : Algebraic geometry and complex geometry / Géométrie algébrique et géométrie complexe

In diophantine geometry, the Batyrev-Manin-Peyre conjecture originally concerns rational points on Fano varieties. It describes the asymptotic behaviour of the number of rational points of bounded height, when the bound becomes arbitrary large. A geometric analogue of this conjecture deals with the asymptotic behaviour of the moduli space of rational curves on a complex Fano variety, when the 'degree' of the curves 'goes to infinity'. Various examples support the claim that, after renormalisation in a relevant ring of motivic integration, the class of this moduli space may converge to a constant which has an interpretation as a motivic Euler product. In this talk, we will state this motivic version of the Batyrev-Manin-Peyre conjecture and give some examples for which it is known to hold : projective space, more generally toric varieties, and equivariant compactifications of vector spaces. In a second part we will introduce the notion of equidistribution of curves and show that it opens a path to new types of results.

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Citation data

  • DOI 10.24350/CIRM.V.19983203
  • Cite this video Faisant Loïs (11/28/22). Asymptotic behaviour of rational curves. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19983203
  • URL https://dx.doi.org/10.24350/CIRM.V.19983203


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