Counting points on a given degree via the height zeta function

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 $X$ 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 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 shed light on recent counter-examples to the original version of Manin's conjecture and on its latest refinements.