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Semi-stable models for curves with cusps. (English) Zbl 1023.14009
Summary: Let $$R$$ be a complete discrete valuation ring with residue characteristic zero, and let $$X$$ be an integral regular flat curve over $$R$$ with smooth generic fiber. Assume that the special fiber of $$X$$ is smooth outside a single point where it has a cusp as singularity. We explicitly determine the structure of the minimal semi-stable model of X. In particular, we give an algebraic proof for the fact that the special fiber of any semi-stable model of $$X$$ is treelike. This is equivalent to the finiteness of the monodromy of $$X$$ over R. These two results were obtained in the 1970’s by Lê Dung Tráng [Compos. Math. 25, 281-321 (1972; Zbl 0245.14003)] and A. Durfee [Invent. Math. 28, 231-241 (1975; Zbl 0278.14010)] using analytic methods.
##### MSC:
 14H20 Singularities of curves, local rings 13F30 Valuation rings 14H10 Families, moduli of curves (algebraic) 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 05C05 Trees
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