Counting points on a given degree via the height zeta function

Kevin Destagnol

Let \(X=\mbox{Sym}^d \mathbf{P}^n:=\mathbf{P}^n \times \cdots \times \mathbf{P}^n/\mathfrak{S}_d\) where the symmetric \(d\)-group acts by permuting the \(d\) copies of \(\mathbf{P}^n\). Manin's conjecture gives a precise prediction for the number of rational points on \(X\) of bounded height in terms of geometric invariants of a resolution of \(X\) and the study of Manin's conjecture for \(X\) can be derived from the geometry of numbers in the cases \(n>d\) and for \(n=d=2\). In this talk, I will explain how one can use the fact that \(\mathbf{P}^n\) is an equivariant compactification of an algebraic group and the height zeta function machinery developed by Chambert-Loir and Tschinkel in order to study the rational points of bounded height on \(X\) in new cases that are not covered by the geometry of numbers techniques. This might in particular be an interesting testing ground for the latest refinements of Manin's conjecture.