The global log canonical threshold (or Tian's alpha-invariant) plays an
important role in the geometry of Fano varieties. Tian showed that Fano
manifolds with big alpha-invariant can be equipped with a Kahler-Einstein
metric. In recent years Donaldson drafted a programme to determine when a
smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that
the existence of such a metric is equivalent to X being K-stable, an
algebraic-geometric property. A crucial step in Donaldson's programme consists
on finding a Kahler-Einstein metric with edge singularities of small angle
along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed
that a dynamic version of the alpha-invariant could be used to find such
metrics.
The global log canonical threshold measures how anticanonical pairs fail to
be log canonical. In this thesis we compute the global log canonical threshold
of del Pezzo surfaces in various settings. We extend Cheltsov's computation of
the global log canonical threshold of complex del Pezzo surfaces to
non-singular del Pezzo surfaces over a ground field which is algebraically
closed and has arbitrary characteristic. Then we study which anticanonical
pairs fail to be log canonical, giving a classification of very singular
anticanonical pairs for del Pezzo surfaces of small degree. We conjecture under
which circumstances such a classification is plausible for an arbitrary Fano
variety and derive consequences. As an application, we compute the dynamic
alpha-invariant on smooth del Pezzo surfaces of small degree with any smooth
elliptic curve as boundary.
The main result of this thesis is a computation of the dynamic
alpha-invariant on all smooth del Pezzo surfaces with boundary any smooth
elliptic curve C. The values of the alpha-invariant depend on the choice of C.
We apply our computation to find Kahler-Einstein metrics with edge
singularities