Let $Y$ be a complex Enriques surface whose universal cover $X$ is birational to a general quartic Hessian surface. Using the result on the automorphism group of $X$ due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of $Y$. The list of elliptic fibrations on $Y$ and the list of combinations of rational double points that can appear on a surface birational to $Y$ are presented. As an application, a set of generators of the automorphism group of the generic Enriques surface is calculated explicitly.

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

If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$ , $q(X)=0$ , $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of  $X$ . We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$ , so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$ . We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.

We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$ , $16\mathsf{A}_{2}$ , $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$ .

Mori dream spaces form a large example class of algebraic varieties, comprising the well-known toric varieties. We provide a first software package for the explicit treatment of Mori dream spaces and demonstrate its use by presenting basic sample computations. The software package is accompanied by a Cox ring database which delivers defining data for Cox rings and Mori dream spaces in a suitable format. As an application of the package, we determine the common Cox ring for the symplectic resolutions of a certain quotient singularity investigated by Bellamy–Schedler and Donten-Bury–Wiśniewski.

We compute the global log canonical thresholds of quasi-smooth well-formed complete intersection log del Pezzo surfaces of amplitude 1 in weighted projective spaces. As a corollary we show the existence of orbifold Kähler—Einstein metrics on many of them.

We present an improved algorithm for the computation of Zariski chambers on algebraic surfaces. The new algorithm significantly outperforms the currently available method and therefore allows us to treat surfaces of high Picard number, where huge numbers of chambers occur. As an application, we efficiently compute the number of chambers supported by the lines on the Segre–Schur quartic.

Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. Upper bounds on the volume and on the number of boundary lattice points of these polygons are derived in terms of the index ℓ. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all ℓ≤16 is obtained.

If C is a curve of genus 2 defined over a field k and J is its Jacobian, then we can associate a hypersurface K in ℙ3 to J, called the Kummer surface of J. Flynn has made this construction explicit in the case when the characteristic of k is not 2 and C is given by a simplified equation. He has also given explicit versions of several maps defined on the Kummer surface and shown how to perform arithmetic on J using these maps. In this paper we generalize these results to the case of arbitrary characteristic.