Symmetries of Del Pezzo Surfaces

For each del Pezzo surface X of degree d, the action of the automorphism group Aut(X) on the exceptional curves of X induces a map ρ : Aut(X) → W(Rd), where W(Rd) is the Weyl group of a root system Rd dependent on the degree of X. The image of ρ is well defined up to conjugacy in W(Rd), so we say that a group G acts by automorphisms on a del Pezzo surface X of degree d if a representative of the conjugacy class of G in W(Rd) is contained in ρ(Aut(X)). The purpose of this thesis is to determine, for each field k of characteristic zero, and for each degree d ≥ 3, the subgroups of W(Rd) that act by automorphisms on a del Pezzo surface of degree d over k. We also provide explicit equations for del Pezzo surfaces of degrees 3 and 4 that admit these group actions, and we determine which of these surfaces are k-rational, stably k-rational, or k-unirational. If a group G acts on a k-rational del Pezzo surface, there is an associated embedding ι : G → Cr2(k) into the plane Cremona group over k. In this way, we make progress toward a classification of the finite subgroups of Cr2(k) over any field of characteristic zero.