Density of rational points on a family of del Pezzo surface of degree 1

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? If not, is it dense with respect to the Zariski topology? Del Pezzo surfaces are classified by their degrees d (an integer between 1 and 9). Manin and various authors proved that for all del Pezzo surfaces of degree >1 is dense provided that the surface has a k-rational point (that lies outside a specific subset of the surface for d=2). For d=1, the del Pezzo surface always has a rational point. However, we don't know it the set of rational points is Zariski-dense. In this talk, I present a result that is joint with Rosa Winter in which we prove the density of rational points for a specific family of del Pezzo surfaces of degree 1 over k.

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