Algebraic points of bounded height on a surface

In this article, we study the asymptotic cardinality of the set of algebraic points of fixed degree and bounded height of a surface defined over a number field, when the bound on the height tends to infinity. In particular, we show that this can be connected to the Batyrev-Manin-Peyre conjecture, i.e. the case of rational points, on some punctual Hilbert scheme. Our study shows that these associated Hilbert schemes provide, under certain conditions, new counterexamples to the Batyrev-Manin-Peyre conjecture. However, in the cases of P-1 x P-1 and P-2 detailed in this article, the associated Hilbert schemes satisfy a slightly weaker version of the Batyrev-Manin-Peyre conjecture.