We show that complete intersection log del Pezzo surfaces with amplitude one in weighted projective spaces are uniformly $K$-stable. As a result, they admit an orbifold Kähler–Einstein metric.

We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.

We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm to classify all of them.

Let $S\subseteq \mathbb{P}^{d}$ be an anticanonically embedded surface of degree $d\geq 3$ . In this note, we classify stable Ulrich bundles on $S$ of rank two. We also study their moduli spaces.

Let S be a degree six del Pezzo surface over an arbitrary field F. Motivated by the first author's classification of all such S up to isomorphism [3] in terms of a separable F-algebra B×Q×F, and by his K-theory isomorphism Kn(S) ≅ Kn(B×Q×F) for n ≥ 0, we prove an equivalence of derived categories

where A is an explicitly given finite dimensional F-algebra whose semisimple part is B×Q×F.

We classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein canonical del Pezzo surface.

We classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a ℚ-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in the 3-fold minimal model program and also in moduli problems.