δ-invariants of Du Val del Pezzo surfaces of degree 2

The δ δ -invariant is a central algebro-geometric quantity that reflects the stability properties of a Fano variety and plays an important role in detecting the existence of Kähler–Einstein metrics. In recent years, explicit computations of δ δ -invariants have become essential in the study of K -stability of Fano varieties, particularly in applications of the Abban–Zhuang method, which reduces higher-dimensional problems to the study of lower-dimensional subvarieties. Del Pezzo surfaces with Du Val singularities are natural objects in this context. In this article, we compute the δ δ -invariants of all Du Val del Pezzo surfaces of degree 2. We show that, for such a surface, the δ δ -invariant is uniquely determined by three geometric features: its degree, the number of lines it contains, and the types of its singularities, and we compute the precise values explicitly. As part of the argument, we also give an explicit description of the Mori cones of Du Val del Pezzo surfaces of degree 2. These computations are particularly useful in applications of the Abban–Zhuang method to the K -stability of higher-dimensional Fano varieties. In particular, they provide the necessary computations of δ δ -invariants for proofs of K -stability for Fano threefolds in the families 1.12, 2.2, and 2.3 of the Mori–Mukai classification.