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Simultaneous resolution of singularities. (English) Zbl 0974.14010
Let \(k\) be a field of characteristic zero, \(L/k\) an \(n\)-dimensional algebraic function field, \(K\) a finite algebraic extension of \(L\), \(\nu\) a zero-dimensional valuation of \(K/k\), and \((R,M)\) a regular local ring, essentially of finite type over \(k\) with quotient field \(K\) and ground field \(k\) such that \(\nu\) has center \(M\) in \(R\). Then for some sequence of monoidal transforms \(R\to R^*\) along \(\nu\), there exists a local domain \(S^*\), essentially of finite type over \(k\) with quotient field \(L\) and ground field \(k\) lying below \(R^*\). When \(n=2\) this is stated by S. S. Abhyankar in theorem 4.8 of his book “Ramification theoretic methods in algebraic geometry” [Ann. Math. Stud. No. 43 (1959; Zbl 0101.38201)]. The above result is a kind of simultaneous resolution of singularities; other forms are also included.

MSC:
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
13A18 Valuations and their generalizations for commutative rings
14B05 Singularities in algebraic geometry
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