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Department of Mathematical Sciences

# Seminar details

## Arithmetic Study Group: Rational curves on cubic hypersurfaces over $\mathbb{F}_q$

Presented by Adelina Manzateanu, U Bristol

5 December 2017 14:05 in CM219

Using a version of the Hardy -- Littlewood circle method over
$\mathbb{F}_q(t)$, one can count $\mathbb{F}_q(t)$-points of bounded degree
on a smooth cubic hypersurface $X \subset \mathbb{P}^{n-1}_{\mathbb{F}_q}$.
Moreover, there is a correspondence between the number of
$\mathbb{F}_q(t)$-points of bounded height and the number of
$\mathbb{F}_q$-points on the moduli space
$\text{Mor}_d(\mathbb{P}^1_{\mathbb{F}_q}, X)$, 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 $\mathbb{F}_q$ 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.