On proximity relations for valuations dominating a two-dimensional regular local ring
Singularities of plane curves can be studied using quadratic transformations of regular local rings of dimension 2. This is a classical topic in commutative algebra and algebraic geometry. Iterated quadratic transformations applied to a given regular local ring of dimension 2 yield an infinite sequence of regular local rings, \(R_0\subset R_1\subset R_2\dots\), whose union is a valuation ring. The relationship between the rings in the sequence can be described by the so-called proximity relation. The authors introduce a sequence of rational numbers as a new numerical invariant of the proximity relation and call it the proximity sequence. It is shown that the proximity sequence encodes the same information as the refined proximity matrix. The sequences that arise as proximity sequences are characterized, and their connection with curve singularities is discussed.
quadratic transformations of regular local rings
scientific article; zbMATH DE number 1419790