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.

中文翻译：

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Stiefel-Whitney曲线覆盖类别

令\（D \）为所有残基场均不等于2的特征的Dedekind方案。对于仅具有奇数分枝的每个驯服度\（C \ to D \），我们将第二个同调中的第二个Stiefel-Whitney类关联具有与（（D \））相关联的某个驯服双曲面\（[D] \）的mod 2系数。然后，该类与第二分支的Stiefel-Whitney类的后退有关，即分枝除数的一半的线束的前推。这（间接地）表明，我们的Stiefel-Whitney类是Esnault，Kahn和Viehweg在“奇数分支的覆盖和Stiefel-Whitney类”中考虑的同调类的总和。也许更重要的是，在代数闭合场上的曲线适当且平滑的情况下，