We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups and describe the connected components of their automorphisms groups. Everywhere below, the number n 0 refers to the corresponding surface in Table 3 in Section 8. As a consequence of our classification we have the following. Corollary 1.2. Let X be a Du Val del Pezzo surface. Then the group Aut(X) is not reductive if and only if X is one of the 23 surfaces 7 Del Pezzo surfaces with Du Val

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... s Let X be a Du Val del Pezzo surface with d := K 2 X . Then d is known as the degree of the surface X. Let µ : X → X be the minimal resolution of singularities. Then so that X is a weak del Pezzo surface, that is, the anticanonical divisor −K X is nef and big. By the Noether formula, d = 10 − ρ(X) 9, and by the genus formula, every irreducible curve on X with negative self-intersection number is either a (−1)-or a (−2)-curve. Moreover, one of the following holds (see [Bre80, HW81] ):