Reduction of Sections

Abstract

Let S = Spec(R) be the spectrum of an excellent henselian discrete valuation ring with generic point Spec(k) and closed point $\mathrm{Spec}(\mathbb{F})$ with $\mathbb{F}$ a perfect field of characteristic p ≥ 0. For a geometrically connected, proper variety X ∕ k, which is the generic fibre $j : X \subseteq \mathcal{X}$ of a proper flat model $f : \mathcal{X} \rightarrow S$ with ${f}_{{\ast}}{\mathcal{O}}_{\mathcal{X}} = {\mathcal{O}}_{S}$, we have a specialisation map

$$X(k) = \mathcal{X}(R) \rightarrow Y (\mathbb{F}),$$

where $Y = {\mathcal{X}}_{\mathbb{F},\mathrm{red}}$ is the underlying reduced subscheme of the special fibre ${\mathcal{X}}_{\mathbb{F}}$.The corresponding structure for sections of ${\pi }_{{}_{1}}(X/k)$ distinguishes between sections that specialise like rational points, namely the sections with vanishing ramification, see Definition 82, and the ramified sections. In the case of curves, the ramication must be purely wild, see Proposition 91, but the case of wild ramification has not been excluded yet.