Toric Degenerations of Polynomial Ideals and Geometric Localization of Fans

Toric degenerations of polynomial ideals occur if one allows certain partial term orders in the theory of Gröbner bases. We prove that the collection of toric degenerations of an ideal can be monotonically embedded into the complex closure of the Gröbner fan. A process of geometric localization on the Gröbner fan of the ideal yields the Gröbner fans of the toric degenerations.