A geometric determinant method and geometric dimension growth

We study a geometric version of the dimension growth conjecture. While it is closely related in spirit to themes arising in geometric Manin's conjecture, it applies in greater generality and provides more uniform bounds. For an irreducible projective variety $X$ defined over $\mathbb{C}(t)$, the set $X(b)$ of $\mathbb{C}(t)$-rational points on $X$ of degree less than $b$ has a natural structure of an algebraic variety over $\mathbb{C}$. We study the dimension and irreducibility of $X(b)$ when $X$ has degree $d \ge 2$, and obtain a geometric analogue of the classical dimension growth conjecture, namely that $\dim X(b) \le b\dim X$ for every $b \ge 1$. In particular, when $X$ is defined over $\mathbb{C}$, this provides uniform bounds on the dimension of the space of degree $b$ rational curves on $X$. We also develop a geometric version of Heath-Brown's $p$-adic determinant method for varieties defined over $\mathbb{C}(t)$. This allows us to show that as soon as $d \ge 6$, the number of irreducible components of $X(b)$ of dimension $b\dim X$ is bounded by a polynomial in $d$ which is independent of $b$. As a further application, we obtain an analogue of the Bombieri--Pila theorem for affine curves, as well as a corresponding result for projective curves.