| Résume | The Hardy–Littlewood circle method is a well-known technique of analytic number theory that has successfully solved several major number theory problems. In particular, it has been instrumental in the study of rational points on hypersurfaces of low degree. More recently, a version of the method over function fields, combined with spreading out techniques, has led to new results about the geometry of moduli spaces of rational curves on hypersurfaces of low degree. In this talk I will show how to implement a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties, and explain how this leads to a more precise description of the geometry of the above moduli spaces. This is joint work with Tim Browning.
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