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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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