Arf closure relative to a divisorial valuation and transversal curves

Abstract

A subring of a discrete valuation ring Rv is called an Rv-Arf ring if it satisfies a certain condition first considered by C. Arf some forty-five years ago. Let A be any subring of Rv and let I be an ideal of A. The operation of the blow up of A at I along the valuation v, and the Arf closure of A with respect to v are explained. Their algebraic structure and main properties are studied. It is shown that, under some assumptions, the family of Arf subrings of Rv satisfies the ascending chain condition. The relation between Arf subrings of Rv and strictly closed subrings of Rv is studied. In the last section the geometrical interpretation of the Arf closure is given.