  Let k be an algebraically closed field of characteristic 0 and let K*/K be a
finite extension of algebraic function fields of transcendence degree 2 over k.
Let v* be a k-valuation of K* with valuation ring V* and let v be the
restriction of v* to K. Suppose R --> S is an extension of algebraic regular
local rings with quotient fields K and K*, respectively, such that V* dominates
S and S dominates R. We prove that there exist sequences of quadratic
transforms R --> R' and S --> S' along v* such that S' dominates R' and the map
between generating sequences of v and v* in R' and S', respectively, has a
toroidal structure. Our result extends the Strong Monomialization theorem of
Cutkosky and Piltant.
