Stable reduction of finite covers of curves

By Qing Liu. Compositio Mathematica, 2004

49 citations recorded by the source corpus.

Let K be the function field of a connected regular scheme S of dimension 1, and let $f : X\to Y$ be a finite cover of projective smooth and geometrically connected curves over K with $g(X)\ge 2$. Suppose that f can be extended to a finite cover ${\mathcal X} \to {\mathcal Y}$ of semi-stable models over S (it is known that this is always possible up to finite separable extension of K). Then there exists a unique minimal such cover. This gives a canonical way to extend $X\to Y$ to a finite cover of semi-stable models over S.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4a2ed0ceb3f2b5f02afced07bab7276cdcb1768d

Citations in the corpus

• Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta (2013, unknown): “The morphism X ′ 1 → X1 is called the stable marked hull of X ′ X in [34].”
• Arithmetic Geometry of Toric Varieties. Metrics, Measures and Heights (2011, unknown): “This follows, for instance, from [Liu06, Corollary 2.8].”