Monomial resolutions of morphisms of algebraic surfaces

Abstract

Dedicated to Robin Hartshorne on the occasion of his sixtieth birthday 1 Introduction.Let k be a perfect field and L/K be a finite separable field extension of onedimensional function fields over k. A classical result (c.f. I.6, [Ha]) states that K (resp. L) has a unique proper and smooth model C (resp. D), and that there is a unique morphism of curves f : D → C inducing the field inclusion K ⊂ L at the generic points of C and D. It has the following properties:(i) f is a finite morphism.(ii) f is monomial on its tamely ramified locus; let β ∈ D be any point, with α := f (β) ∈ C, such that the extension of discrete valuation rings O Y,β /O X,α is tamely ramified. There exists a local-étale ring extension R of O Y,β and regular parameters u of O X,α and x of R such that u = x a for some a prime to the characteristic of k.

Torification and factorization of birational maps

“…This result has been proven in an algorithmic manner by Cutkosky and Piltant [19]. Similar statements can be found in [6].…”

Ramification of valuations

“…Then W and W * are divisorial valuation rings and W * ∩ K = W . The argument in the proof of lemma 2 of [13] now shows that the ramification index e(W * /W ) of W * /W is δ and that there is an inclusion…”

“…(1) f u is a unit in S and f v is not a unit in S. Proof. We review the proof of lemma 2 of [13] and point out the appropriate changes. Let W * := S (x) and W := R (x)∩R .…”

Resolution of Singularities

“…We discuss some of these results in Chapter 9. A generalization of monomialization in characteristic p function fields of algebraic surfaces is obtained in [34] and especially in [35].…”