An arithmetic Hurwitz formula (Q1341244)

Let \(f : X \to Y\) be a tame and finite morphism of degree \(n\) between two semistable curves over the spectrum of a complete discrete valuation ring with algebraically closed residue field. Let \(\underline c_2 (Y)\) denote the number of double points on the special fiber \(Y_k\) of \(Y\). Results of Bloch allow the author to reinterpret \(\underline c_2 (Y)\) as the degree of the push-forward of a localized Chern class. The author expresses the difference \(\underline c_2 (X) - n \underline c_2 (Y)\) in terms of data related to the ramification of the map \(f\). An application to the case of curves of genus 2 is given [see also \textit{Q. Liu}, ``Conducteur et discriminant minimal de courbes de genre 2'', Compos. Math. 94, No. 1, 51-79 (1994)].