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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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