The Quadratic Manin-Peyre conjecture for del Pezzo surfaces

Abstract: Manin-Peyre conjecture, counting point of bounded height on Fano varieties, has been the subject of intense research in the past few decades. We provide a general framework for the Manin-Peyre conjecture for the symmetric square of any del Pezzo surface X, and prove the conjecture for the infinite family of nonsplit quadric surfaces. Previously, there were only two examples in the literature: Sym^2(P^2) and Sym^2(P^1 x P^1). In order to achieve the predicted asymptotic, we show that a type II thin set of a new flavour must be removed. A key tool we develop and that can be applied to further examples is a result for summing multiplicative functions and Euler products over quadratic extensions. To establish our counting result for the specific family of quadric surfaces, we improve existing lattice point counting results in the literature and make crucial use of a novel form of lattice point counting. This work is joint with Francesca Balestrieri, Kevin Destagnol, Julian Lyczak, and Nick Rome.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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