Stability and Einstein-Kähler Metric of a Quartic del Pezzo Surface

A del Pezzo orbifold S is a compact irreducible complex surface, with only isolated quotient singularities, such that c₁(S) > 0. The self-intersection number c₁(S)² is called its degree, and a smooth S is called a del Pezzo surface. Let Σ be the set of all isomorphism classes of quartic del Pezzo surfaces. In this paper, we study two geometric compactifications Σ ¯ E K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071891/a5dc7e8e-a6e9-40ba-9f3f-b67cf71870a5/content/eq1830.tif"/> and Σ ¯ alg https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071891/a5dc7e8e-a6e9-40ba-9f3f-b67cf71870a5/content/eq1831.tif"/> of Σ and show that they coincide.