Counting points on Hilbert schemes over function fields

Adelina Manzateanu

We consider the Hilbert scheme \(\text{Hilb}^2\mathbb{P}^2\) defined over a global field of positive characteristic \(K\). We give an asymptotic formula for the number of \(K\)-points of bounded height on \(\text{Hilb}^2\mathbb{P}^2\) that supports the thin set version of Manin's conjecture and show that the leading constant agrees with the prediction of Peyre. Moreover, we extend the analogy between the integers and 0-cycles on a variety \(V\) over a finite field to 0-cycles on a variety \(V\) over \(K\) and establish a version of the prime number theorem in the case when \(V=\mathbb{P}^2\).