Quotients of del Pezzo surfaces of degree 1

In the talk we consider rationality of quotients of del Pezzo surfaces of degree 1 by finite subgroups of automorphism group over algebraically nonclosed fields of characteristic 0. For this case we show that there are only five possibilities for groups, such that the quotient is not rational over the ground field: trivial group, cyclic group of order 2, two types of cyclic groups of order 3, and cyclic group of order 6. For all other automorphism group we show that the quotient is rational (if there exists a smooth point defined over the ground field). For the trivial group and the group of order 2 acting minimally the surface an the quotient are always not rational. For three other groups acting minimally we construct all possible examples of rational/nonrational quotients of rational/nonrational surfaces.
At the end of the talk we will discuss different corollaries from the obtained results: fields of invariants, classification of Galois-unirational surfaces, applications to the plane Cremona group, etc.