Simple complete ideals in two-dimensional regular local rings

Abstract

In a two-dimensional regular local ring (R,m), it is known that there exists a unique complete ideal I adjacent to a given simple complete m-primary ideal J from above. In this paper it is shown that there are infinitely many simple complete m-primary ideals adjacent to a given simple complete m-primary ideal J (≠ m) from below whose orders are the same as that of J and that there exists a unique complete m-primary ideal adjacent to J from below whose order is one bigger than that of J. We also show that these are all the complete ideals adjacent to J from below. It is known that there is a unique prime divisor w and a unique infinitely near point S of R associated to a given simple complete m-primary ideal J. As a corollary of the main theorem, we obtain one-to-one correspondences between the set of simple m-primary complete ideals adjacent to J from below, the set of first neighborhood prime divisors of w and the set of first quadratic transformations of S.