Inverse Galois problem for degree‑1 del Pezzo surfaces over finite fields

Determine, for every cyclic conjugacy class in the Weyl group W(E8) and every finite field F_q, whether there exists a del Pezzo surface X of degree 1 over F_q such that the image of the absolute Galois group of F_q acting on the geometric Picard group Pic( X ⊗ F̄_q ) lies in that conjugacy class; equivalently, solve the inverse Galois problem for del Pezzo surfaces of degree 1 over finite fields.

Background

Del Pezzo surfaces of degree d≤6 have their Galois action on the geometric Picard group identified with a subgroup of the Weyl group W(E_{9−d}); in degree 1 this is W(E8), which has 112 cyclic conjugacy classes (types). For finite fields, the inverse Galois problem asks, for each cyclic conjugacy class, whether there exists a del Pezzo surface of the given degree over a fixed finite field whose Frobenius action on Picard realizes that class.

For degrees ≥2 the problem over finite fields has been solved, but for degree 1 it remains incomplete. The present work resolves 85 of the 112 types and determines minimal fields of existence for all 112, yet a full resolution for all types and fields remains open, motivating the explicit formulation of the outstanding inverse Galois problem in degree 1.