Dynamic alpha-invariants of del Pezzo surfaces

For every smooth del Pezzo surface S, smooth curve C ∈ |- K_S| and β ∈ (0, 1], we compute the α-invariant of Tian α(S, (1 - β)C) and prove the existence of Kähler-Einstein metrics on S with edge singularities along C of angle 2πβ for β in certain interval. In particular, we give lower bounds for the invariant R(S, C), introduced by Donaldson as the supremum of all β ∈ (0, 1] for which such a metric exists. The pairs (S, C) considered are strongly asymptotically log del Pezzo surfaces. We study one of the two classes of such pairs for which such metrics are expected to exist for all small β >0.