Space Valuations Are Not Uniquely Determined by Their Centers

Abstract

The sequence of centers and multiplicities for any valuation of the local ring of a point p in an n-dimensional non-singular variety is defined. A Noether formula for valuations is obtained, which leads to some proximity relations between the multiplicities of the valuation in its centers. We will consider a sequence of irreducible subschemes infinitely near p, each one in the first neighborhood of the preceding one, and with the condition that infinitely many of them are free. Then, a valuation with those centers and multiplicities exists which, unlike the plane case, is not unique. To show this, an example of a space valuation, built as the lifting of a plane valuation, is provided.

Key Words:

• Valuation
• Infinitely near schemes
• Sequence of centers
• Noether formula
• Proximity relations

Mathematics Subject Classification:

• 13A18
• 14B05

Notes

^#Communicated by A. Facchini.