Automorphisms of Finite Order on Rational Surfaces: With an appendix by I. Dolgachev

We classify minimal pairs (X, G) for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus XG and the quotient surface Y = X/G as well as the fundamental group of the smooth part of Y. The realization of each pair is included. Moriʹs extremal ray theory and recent results of Alexeev and also Ambro on the existence of good anti-canonical divisors are used.