Let π: X → ℙn be the d-cyclic covering branched along a smooth hypersurface Y ⊂ ℙn of degree d, 3 ≤ d ≤ n. We identify the minimal rational curves on X with d-tangent lines of Y and describe the scheme structure of the variety of minimal rational tangents 𝒞x ⊂ ℙTx(X) at a general point x ∈ X. We also show that the projective isomorphism type of 𝒞x varies in a maximal way as x moves over general points of X.

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

We show that any $n$-dimensional Fano manifold $X$ with $\unicode[STIX]{x1D6FC}(X)=n/(n+1)$ and $n\geqslant 2$ is K-stable, where $\unicode[STIX]{x1D6FC}(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits Kähler–Einstein metrics and the holomorphic automorphism group $\operatorname{Aut}(X)$ of $X$ is finite.

We show that the pair (X, –KX ) is K-unstable for a del Pezzo manifold X of degree 5 with dimension 4 or 5. This disproves a conjecture of Odaka and Okada.

We develop a new non-vanishing theorem for del Pezzo surfaces with quotient singularities.

In this work we provide effective bounds and classification results for rational ℚ-factorial Fano varieties with a complexity-one torus action and Picard number 1 depending on the two invariants dimension and Picard index. This complements earlier work by Hausen et al., where the case of a free divisor class group of rank 1 was treated.