Abstract. We construct the extension of a hyperelliptic K3 surface to a Fano 6-fold with extraordinary properties in moduli. This leads us to a family of surfaces of general type with pg = 1, q = 0, K 2 = 2 and hyperelliptic canonical curve, each of which is a weighted complete intersection inside a Fano 6-fold. Finally, we use these hyperelliptic surfaces to determine an 8-parameter family of Godeaux surfaces with π 1 = Z/2.

Cluster algebras give rise to a class of Gorenstein rings which enjoy a large amount of symmetry. Concentrating on the rank 2 cases, we show how cluster varieties can be used to construct many interesting projective algebraic varieties. Our main application is then to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In particular, for Fano 3-folds in codimension 4 we construct at least one family for 187 of the 206 possible Hilbert polynomials contained in the Graded Ring Database.

We present a list of arithmetically Gorenstein Calabi-Yau threefolds in P 7 and give evidence that this is a complete list. In particular we construct three new families of arithmetically Gorenstein Calabi-Yau threefolds in P 7 for which no mirror construction is known.

We use constructions of surfaces as abelian covers to write down exceptional collections of line bundles of maximal length for every surface X in certain families of surfaces of general type with p g = 0 and K 2 X = 3, 4, 5, 6, 8. We also compute the algebra of derived endomorphisms for an appropriately chosen exceptional collection, and the Hochschild cohomology of the corresponding quasiphantom category. As a consequence, we see that the subcategory generated by the exceptional collection does not vary in the family of surfaces. Finally, we describe the semigroup of effective divisors on each surface, answering a question of Alexeev.

We show that the Kulikov surfaces form a connected component of the moduli space of surfaces of general type with p g = 0 and K 2 = 6. We also give a new description for these surfaces, extending ideas of Inoue. Finally we calculate the bicanonical degree of Kulikov surfaces, and prove that they verify the Bloch conjecture.