More about dicriticals

Abstract

Abstract. This paper attempts to characterize the dicritical set of a rational function which generates a special pencil at a simple point of an algebraic or arithmetical surface.

Existence of dicritical divisors revisited

“…In this paper we shall make further progress in this theory. In particular, in Theorem (8.2) we shall prove a converse of Proposition (3.5) of [Ab11] characterizing the dicritical set of a special pencil on a nonsingular surface; see Section 8 for the statement of this and other related results. In Section 9 we shall initiate the dicritical theory of higher dimensional normal varieties, which may be algebraic or arithmetical, and we shall indicate how this could be used in attacking the higher dimensional jacobian conjecture.…”

“…It was algebracized in [Ab8,Ab9]. The algebraic theory was furthered in [Ab10,Ab11,AbH,AbL]. In this paper we shall make further progress in this theory.…”

“…In Section 9 we shall initiate the dicritical theory of higher dimensional normal varieties, which may be algebraic or arithmetical, and we shall indicate how this could be used in attacking the higher dimensional jacobian conjecture. We shall use the notation and terminology introduced in [Ab3] to [Ab9] and more specifically in [Ab10,Ab11]. Relevant background material can be found in [Ab1,Ab2,Nag,NoR,Zar].…”

“…Details are in the third paragraph after (2.3) of [Ab11] where you can also find the the definitions of: an element or subset of a nonnull ring S to be integral over an ideal J in a subring of S, the integral closure of J in S, and the reduction of an ideal. Q(R) = j∈N Q j (R) = the set of all two dimensional regular local domains which birationally dominate R, i.e., whose quotient field is L and which dominate R; proof of the second equality in [Ab1].…”

“…is the bijective map given in Appendix 5 of [Zar] which we call the Zariski map. Details are in Section 2 of [Ab11]. a R (z) (resp: b R (z), J R (z), I R (z)) = the numerator (resp: denominator, first asociated, second associated) ideal of z in R. Details are in the fourth paragraph after (2.3) of [Ab11].…”