### Density of rational points on a family of del Pezzo surfaces of degree 1

Heilbronn Number Theory Seminar

26th February 2020, 2:00 pm – 3:00 pm

Fry Building, G.07

Let k be a number field and X an algebraic variety over k. We want to study the set of k-rational points X(k). For example, is X(k) empty? And if not, is it dense with respect to the Zariski topology? Del Pezzo surfaces are surfaces that are classified by their degree d (for d ≥ 3, these are the surfaces of degree d in P^d). For all del Pezzo surfaces of degree ≥ 2 over k, we know that the set of k-rational points is dense provided that the surface has a k-rational point (that lies outside a specific subset of the surface for degree 2).

But for del Pezzo surfaces of degree 1 over k, even though we know that they always contain a k-rational point, we do not know if the set of k-rational points is dense. In this talk I will focus on a result that is joint work with Julie Desjardins, in which we prove that for a specific family of del Pezzo surfaces of degree 1 over k, under a mild condition, the k-rational

points are dense with repsect to the Zariski topology. I will compare this to previous results.

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