Following Serre's work on the solubility of conics, a current problem of interest in the study of Diophantine equations is to count the number of varieties in families that have a rational point.
In this talk, I will give an overview of recent works in this area, before focussing on the particular example of diagonal quadric surfaces parameterised by {Y: wx=yz}. This family was first studied by Browning, Lyczak, and Sarapin, who showed that it exhibits an uncommonly large number of soluble members and attributed this phenomenon to the existence of thin sets on Y. They predicted that the “typical” behaviour should hold outside of this thin set, in the style of modern formulations of the Batyrev--Manin conjectures. In recent work, I have shown that unusual behaviour occurs even with the removal of these thin sets by providing an asymptotic formula for the corresponding counting problem.
Finally, I will outline the character sum methods used to prove this result and introduce an adaptation of the large sieve for quadratic characters