2025 Theses Doctoral

The arithmetic of del Pezzo surfaces and Hilbert schemes of points

Porzio, Morena

Motivated by the Cassels–Swinnerton-Dyer Conjecture for cubic surfaces, this thesis investigates the stable birational class of Hilbⁿ_𝑋 , the Hilbert scheme of length 𝑛 closed subschemes on a given surface 𝑋. The primary focus is to determine for which pairs of positive integers (𝑛,𝑛′) the varieties Hilbⁿ_𝑋 and Hilbⁿ′_𝑋 are stably birational, specifically when 𝑋 is a surface with irregularity 𝑞(𝑋) = 0.

After establishing general results for such surfaces, the study narrows its scope to geometrically rational surfaces. In this case, it is shown that, among the Hilbⁿ_𝑋’s, there exist only finitely many stable birational classes.

A corollary of this finding is the rationality of the motivic zeta function 𝜁_mot (𝑋, 𝑡) in 𝐾₀ (Var/𝑘)/([A¹_𝑘]) [[𝑡]] over fields of characteristic zero.
Returning to cubic surfaces, the thesis further examines the stable birational types of Hilbⁿ_𝑋 both asymptotically and for small values of 𝑛.