Geometry of an "isotrivial" del Pezzo surface of degree 1

Julie Desjardins

The elliptic surface \(E :y^2=x^3+AT^6+B\) (\(A\),\(B\) rational) is a canonical example where both (1) the density isn't yet proven with geometric means, (2) there are many cases where no (even conditional) result can be deduced from the study of the root number of the fibers ("= parity of the rank" under a weaker Birch Swinnerton-Dyer conjecture). Moreover, the blow down of the section at infinity of this surface gives a del Pezzo surface of degree 1. In this talk, I will speak about a joint work with Bartosz Naskręcki in which we describe the beautiful geometry under this particular type of surface : by describing a base of the Mordell-Weil group, we give an algorithm (with input A and B) allowing to determine the generic rank of the elliptic surface (between 0 and 3). When the rank is non-zero, this proves the Zariski-density of rational points of del Pezzo surface of degree 1 : \(y^2=x^3+Am^6+Bn^6\).