Inversion and Invariance of Characteristic Terms: Part I

Dicritical divisors and Jacobian problem

“…By the first equation of (•) we get The above two theorems generalize the results of [4] to include the case of transcendentality gaps; for details see [16]. The above two theorems can also be used to prove the fundamental property of dicritical divisors, stated in (6.2), for polynomial rings over characteristic zero fields.…”

“…But, although my new paper [16], which connects my 1967 paper [4] to dicritical divisors, exceeded sixty pages and in spite of strenuous attempts, I was stuck in zero characteristic and failed to extend the results to prime characteristic or to the arithmetic case.…”

“…In the rest of the paper I shall give brief expositions of the papers [16,18,17] which I just finished writing.…”

Pillars and towers of quadratic transformations

“…If A is a local domain, then, in view of Krull-Akizuki, D(A, J) is a finite set, i.e., |D(A, J)| < ∞, where | | denotes cardinality; in the case when J is a pencil (as defined below) in a two dimensional regular local domain R = A, this is discussed in the first paragraph of (5.6)( † * ) of [Ab5]; in the general case, it suffices to note that a nonzero ideal in a Noetherian domain is contained in at most a finite number of height one prime ideals. If A is a positive dimensional local domain, then by a QDT = Quadratic Transform of A we mean a member of W(A, M (A)) Δ ; by a 0-th QDT of A we mean A itself, by a first QDT of A we mean a QDT of A,.…”

More about dicriticals

“…The study of dicritical divisors started in Section 5 of [Ab5] was continued in [AbH] and [AbL]. The main results of these two papers will be restated as Propositions 2.1 and 2.2 of Section 2.…”