Valuations Compatible with a Projection

Abstract

Given an N-dimensional germ of analytic hypersurface $\mathcal{H}$, a finite projection π : $\mathcal{H} \to \mathbb{C}^N$ and a valuation v on the ring of convergent series in N variables, we study the valuations on the ring $\mathcal{O}_\mathcal{H}$ that extend π*v. All these valuations are described when v is a monomial valuation whose weight vector is not orthogonal to any of the faces of the Newton Polyhedron of the discriminant of the projection π. This description is done in terms of the Puiseux parameterizations of $\mathcal{H}$ with exponents in a cone.

Supported by grant IN105806 from PAPIIT, UNAM. The research was carried out while the author was in Sao Carlos profiting from a DTI fellowship given by IM-AGIMB processo 381461/2004-1 (NV).