An abelian cover is a finite morphism X → Y of varieties which is the quotient map for a generically faithful action of a finite abelian group G. Abelian covers with Y smooth and X normal were studied in [R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191-213; MR 1103912(92g:14012)]. Here we study the non-normal case, assuming that X and Y are S 2 varieties that have at worst normal crossings outside a subset of codimension greater than or equal to two. Special attention is paid to the case of Z r 2 -covers of surfaces, which is used in [V. Alexeev and R. Pardini,

Abstract. We prove the so-called Severi inequality, stating that the invariants of a minimal smooth complex projective surface of maximal Albanese dimension satisfy:

We classify log-canonical pairs (X, ∆) of dimension two with KX +∆ an ample Cartier divisor with (KX + ∆) 2 = 1, giving some applications to stable surfaces with K 2 = 1. A rough classification is also given in the case ∆ = 0.

In this paper we consider Gorenstein stable surfaces with KX2=1 and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with KX2=1; for pg=2 this leads to a rough stratification of the moduli space. Explicit non‐Gorenstein examples show that we need further techniques to understand all possible degenerations.

A minimal surface of general type with p g (S) = 0 satisfies 1 ≤ K 2 ≤ 9 and it is known that the image of the bicanonical map ϕ is a surface for K 2 S ≥ 2, whilst for K 2 S ≥ 5, the bicanonical map is always a morphism. In this paper it is shown that ϕ is birational if K 2 S = 9 and that the degree of ϕ is at most 2 if K 2 S = 7 or K 2 S = 8. By presenting two examples of surfaces S with K 2 S = 7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K 2 S = 8 is, to our knowledge, a new example of a surface of general type with p g = 0.The degree of ϕ is also calculated for two other known surfaces of general type with p g = 0, K 2 S = 8. In both cases the bicanonical map turns out to be birational.