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We use the techniques of birational algebraic geometry and some combinatorialarguments related to weighted trees to study the structure of resolutions ofcompactifications of hypothetical counterexamples to the two-dimensionalJacobian Conjecture. We obtain especially detailed results for the Steindecomposition of such maps. We hope that our approach will inspire morebirational geometers to seriously work on the Jacobian Conjecture.

On Resolution of Compactifications of Unramified Planar Self-maps