Title: On the inverse Galois problem for del Pezzo surfaces of degree 1
Abstract: Del Pezzo surfaces over a field are nice (= smooth projective geometrically integral) surfaces with an ample anticanonical sheaf. The self-intersection number of the canonical sheaf is called the degree of the del Pezzo surface and is always an integer between 1 and 9. There is a natural action of the absolute Galois group of the ground field on the geometric Picard group of the surface. This action preserves the intersection form and the canonical sheaf. The geometric Picard group depends only on the degree of the surface. The group of automorphisms of the geometric Picard group preserving the intersection form and the canonical sheaf is isomorphic to the Weyl group of a certain root system. The image of the Galois group in the Weyl group is well-defined up to conjugation.
In general, the inverse Galois problem for del Pezzo surfaces (of a fixed degree and ground field) asks which conjugacy classes of the Weyl group are actually realized as the image of the Galois group for a del Pezzo surface.
In the talk, I will focus on the case where the degree is 1 and the ground field is finite. In this case, the corresponding Weyl group is W(E_8), which has 112 cyclic conjugacy classes. In particular, I will explain an algorithmic approach to the problem using the description of del Pezzo surfaces as blow-ups of the projective plane and as hypersurfaces of degree 6 in the weighted projective space P(1,1,2,3), yielding a complete solution for 85 of the 112 classes.