Arkiv för Matematik

Let $D$ be a Dedekind scheme with the characteristic of all residue fields not equal to 2. To every tame cover $C\to D$ with only odd ramification we associate a second Stiefel-Whitney class in the second cohomology with mod 2 coefficients of a certain tame orbicurve $[D]$ associated to $D$ . This class is then related to the pull-back of the second Stiefel-Whitney class of the push-forward of the line bundle of half of the ramification divisor. This shows (indirectly) that our Stiefel-Whitney class is the pull-back of a sum of cohomology classes considered by Esnault, Kahn and Viehweg in ‘Coverings with odd ramification and Stiefel-Whitney classes’. Perhaps more importantly, in the case of a proper and smooth curve over an algebraically closed field, our Stiefel-Whitney class is shown to be the pull-back of an invariant considered by Serre in ‘Revêtements à ramification impaire et thêta-caractéristiques’, and in this case our arguments give a new proof of the main result of that article.

LetD be a domain inR 2 whose complement is contained in a pair of rays leaving the origin. That is,D contains two sectors whose base angles sum to 2π. We use balayage to give an integral test that determines if the origin splits into exactly two minimal Martin boundary points, one approached through each sector. This test is related to other integral tests due to Benedicks and Chevallier, the former in the special case of a Denjoy domain. We then generalise our test, replacing the pair of rays by an arbitrary number.

In the paper [B2] Baernstein constructs a simply connected domain Θ in the plane for which the conformal mappingf of Θ into the unit disc Δ satisfies $$\int_{R \cap \Omega } {|f'(z)|} ^p |dz| = \infty $$ for somep∈(1 2) where R is the real line This gives a counterexample to a conjecture stating that for any simply connected domain Θ in the plane all the above integrals are finite for any 1