Title: On a geometric dimension growth conjecture

Abstract: Let X be an integral projective variety of degree at least 2 defined over Q, and let B>0 an integer. The dimension growth conjecture, now proven in almost all cases following works of Browning, Heath-Brown and Salberger, provides a certain uniform upper bound on the number of rational points of height at most B lying on X.

Shifting to the geometric setting (where X may be defined over C(t)), the collection of C(t)-rational points lying on X of degree at most B naturally forms an algebraic variety, which we denote by X(B). In current work, we uniformly bound the dimension and, when the degree of X is at least 6, the number of irreducible components of X(B) of largest possible dimension, obtaining a geometric analogue of the dimension growth conjecture.

In order to do so, we develop a geometric version of Heath-Brown's determinant method which is of independent interest, and use results on rational points on curves over function fields. 

Joint work with Tijs Buggenhout and Floris Vermeulen.

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