Exceptional del Pezzo hypersurfaces

TL;DR

This paper calculates the global log canonical thresholds for a broad class of quasismooth del Pezzo hypersurfaces in weighted projective spaces, leading to new existence results for orbifold Kähler--Einstein metrics and classifications of exceptional cases.

Contribution

It provides explicit computations of thresholds, establishes existence of Kähler--Einstein metrics, and classifies exceptional hypersurfaces in weighted projective spaces.

Findings

Computed thresholds for many hypersurfaces

Established existence of orbifold Kähler--Einstein metrics

Classified exceptional and weakly exceptional hypersurfaces

Abstract

We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. As a corollary we obtain the existence of orbifold K\"ahler--Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$.