# 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 : \(y^2=x^3+Am^6+Bn^6\).