Stable modification of relative curves

2007/07/26 by Michael Temkin, Temkin, Michael · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0707.3953

openalex publication_date 2007/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize theorems of Deligne-Mumford and de Jong on semi-stable modifications of families of proper curves. The main result states that after a generically étale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fibers have no certain exceptional components. The main step of our proof is uniformization of one-dimensional extensions of valued fields. Riemann-Zariski spaces are then used to obtain the result over any integral base.

Citations

• Smoothness, Semistability, and Toroidal Geometry

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• δ-lifting and 1-dimensional analytic fields
• Continuity of the radius of convergence of differential equations on\n p-adic analytic curves

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