On Toric Resolution of Quasi-ordinary Surface Germs
by
Pedro González Pérez
Univ. de La Laguna - École Normale Supèrieure

A quasi-ordinary surface germ embedded in (C³, 0) is one wich can be given in suitable local coordinates by the vanishing of a polynomial f in C { X, Y } [Z] such that the discriminant \Delta_Z f of f with respect to Z is of the form:

\DeltaZ f = Xk Yl \epsilon(X, Y), with \epsilon(0, 0) =/= 0.

The roots \zeta_i of f, as a polynomial in Z, are fractional power series in the ring C { X^1/n , Y^1/n } wich are conjugate with respect to the action of pairs of n^th-roots of unity. The difference of two distinct conjugates of a root \zeta of f is of the form X^\lambda Y^\mu \beta where \beta is a unit in C {X^1/n , Y^1/n } , and the fractional monomials X^\lambda Y^\mu so obtained are called the characteristic monomials of \zeta.

We show how to construct an embedded resolution of the surface germ f = 0 wich depends only on the characteristic monomials. This resolution involves toric morphisms adapted to the Newton polyhedron of f=0, changes of coordinates and ``translations''. This method generalises the analogous result for plane branches.

Date received: April 22, 1999