A Geometric Approach to the Two-dimensional Jacobian Conjecture

Suppose f(x, y), g(x, y) are two polynomials with complex coefficients. The classical Jacobian Conjecture (due to Keller) asserts the following. Conjecture. (Jacobian Conjecture in dimension two) If the Jacobian of the pair (f, g) is a non-zero constant, then the map (x, y) 7→ (f(x, y), g(x, y)) is invertible. Note that the opposite is clearly true, because the Jacobian of any polynomial map is a polynomial, and, when the map is invertible, it must have no zeroes, so it is a constant. The Jacobian Conjecture and its generalizations received considerable attention in the past, see [3]. It is notorious for its subtlety, having produced a substantial number of wrong ”proofs”, by respectable mathematicians. From the point of view of a birational geometer, the most natural approach to the two-dimensional Jacobian Conjecture is the following. Suppose a counterexample exists. It gives a rational map from P 2 to P . After a sequence of blowups of points, we can get a surface X with two maps: π : X → P 2 (projection onto the origin P ) and φ : X → P 2 (the lift of an original rational map). Note that X contains a Zariski open subset isomorphic to A and its complement, π∗((∞)), is a tree of smooth rational curves. We will call these curves exceptional, or curves at infinity. The structure of this tree is easy to understand inductively, as it is built from a single curve (∞) on P 2 by a sequence of two operations: blowing up a point on one of the curves of blowing up a point of intersection of two curves. However, a non-inductive description is probably impossible, which is the first difficulty in this approach. Another difficulty comes from the fact that the exceptional curves on X may behave very differently with respect to the map φ. More precisely, there are four types of curves E. type 1) φ(E) = (∞)