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Wild quotients of products of curves. (English) Zbl 1401.14023

Let \(k\) be an algebraically closed field of characteristic \(p>0\) and \(X_1, X_2\) be two smooth proper connected curves. Let \(\sigma_i: X_i\to X_i\) be an automorphism of order \(p\) and denote by \(\sigma\) the automorphism \(\sigma_1\times \sigma_2: X_1\times X_2\to X_1\times X_2=:Y\). It is proved that the graph of the resolution of any singularity of \(Y/\langle\sigma \rangle\) is a star-shaped graph with three terminal chains when \(X_2\) is an ordinary curve of positive genus. The intersection matrix of the resolution has determinant \(\pm p^2\). The singularity is rational. It is proved that for any \(s>0\) not divisible by \(p\) there are resolution graphs of wild \(\mathbb{Z}/p\mathbb{Z}\) quotient singularities with one node, \(s+2\) terminal chains and intersection matrix having determinant \(\pm p^{s+1}\).

MSC:

14B05 Singularities in algebraic geometry
14G20 Local ground fields in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14H20 Singularities of curves, local rings
13H15 Multiplicity theory and related topics
14J17 Singularities of surfaces or higher-dimensional varieties
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