Stiefel-Whitney classes of curve covers

Let be a Dedekind scheme with the characteristic of all residue fields not equal to 2. To every tame cover with only odd ramification we associate a second Stiefel-Whitney class in the second cohomology with mod 2 coefficients of a certain tame orbicurve associated to . 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 'Revtements A ramification impaire et thta-caract,ristiques', and in this case our arguments give a new proof of the main result of that article.