Rational curves on cubic hypersurfaces over $\mathbb{F}_q$

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 $\mbox{Mor}_d(\mathbb{P}^{1}_{\mathbb{F}_q}, X)$, 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.