Rational curves on cubic hypersurfaces over Fq

Adelina Mânzățeanu (University of Bristol)

Frank Adams 1,

Using a version of the Hardy - Littlewood circle method over \(F_q(t)\), one can count \(F_q(t)\)-points of bounded degree on a smooth cubic hypersurface \(X \subset P^{n−1}\) over \(F_q\). Moreover, there is a correspondence between the number of \(F_q(t)\)-points of bounded height and the number of \(F_q\)-points on the moduli space which parametrises the rational maps of degree \(d\) on \(X\). In this talk I will give an asymptotic formula for the number of rational curves defined over \(F_q\) on \(X\) passing through two fixed points, one of which does not belong to the Hessian, for \(n \geq 10\), and \(q\) and \(d\) large enough. Further, I will explain how to deduce results regarding the geometry of the space of such curves.
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