EUDML

We give a simpler and more conceptual proof of toroidalization of morphisms of 3-folds to surfaces, over an algebraically closed field of characteristic zero. A toroidalization is obtained by performing sequences of blow ups of nonsingular subvarieties above the domain and range, to make a morphism toroidal. The original proof of toroidalization of morphisms of 3-folds to surfaces is much more complicated.

Local monomialization of transcendental extensions

Steven Dale CUTKOSKY — 2005

Annales de l’institut Fourier

Suppose that $R \subset S$ are regular local rings which are essentially of finite type over a field $k$ of characteristic zero. If $V$ is a valuation ring of the quotient field $K$ of $S$ which dominates $S$ , then we show that there are sequences of monoidal transforms (blow ups of regular primes) $R \to R_{1}$ and $S \to S_{1}$ along $V$ such that $R_{1} \to S_{1}$ is a monomial mapping. It follows that a morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blow ups of nonsingular subvarieties.

Formal prime ideals of infinite value and their algebraic resolution

Steven Dale Cutkosky; Samar ElHitti — 2010

Annales de la faculté des sciences de Toulouse Mathématiques

Suppose that $R$ is a local domain essentially of finite type over a field of characteristic $0$ , and $ν$ a valuation of the quotient field of $R$ which dominates $R$ . The rank of such a valuation often increases upon extending the valuation to a valuation dominating $\hat{R}$ , the completion of $R$ . When the rank of $ν$ is $1$ , Cutkosky and Ghezzi handle this phenomenon by resolving the prime ideal of infinite value, but give an example showing that when the rank is greater than $1$ , there is no natural ideal in $\hat{R}$ that...

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Purity of the branch locus and Lefschetz theorems

Symbolic algebras of monomial primes.

On Fano 3-Folds.

On unique and almost unique factorization of complete ideals II.

A new characterization of rational surface singularities.

A simpler proof of toroidalization of morphisms from 3-folds to surfaces

Local monomialization of transcendental extensions

Formal prime ideals of infinite value and their algebraic resolution