On two invariants of divisorial valuations at infinity

Abstract

The two-dimensional case of the famous Jacobian conjecture of O.-H. Keller asserts that every unramified polynomial self-map of an affine plane is invertible. Many geometric approaches to this conjecture involve divisorial valuations of the field C(x, y), centered outside of the affine plane. Two integer invariants of these valuations naturally appear in this context. In this paper we study these invariants using combinatorics of weighted graphs. In particular, we prove that whenever both invariants are fixed, the corresponding valuations form a finite number of families up to plane automorphisms.

Frameworks for Two-Dimensional Keller Maps

“…In particular, on Y allK labels are non-positive, and all determinant labels are positive (cf. [4] for the definitions). This is how all our frameworks were constructed, by hand.…”

“…Note that the self-intersection numbers change under blowups and contractions, so they are not the invariants of the divisorial valuations defined by the curves at infinity. To get around that, two other labels for these curves were introduced in [3] and [4]: theK label and the determinant label. These labels are invariant under polynomial automorphisms of A 2 .…”

“…Modulo that, the valuations with given labels form finitely many families (cf. [4]). We will not use the determinant labels until section 6, and will define and discuss them there.…”

“…Note that this map is polynomial, and the polynomials have the correct degrees. It has a rather simple Jacobian, a constant multiple of x 4 1 x 12 2 . It is generically 16:1.…”

“…So I will first describe where they came from, and then briefly discuss some open questions that may help us resolve the Jacobian Conjecture in dimension 2. Some of the proofs in this section are only sketched, and certain degree of familiarity with the papers [3] and [4] is highly recommended.…”