On the homology of two-dimensional elimination

Abstract

We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method for their calculation in terms of certain Hilbert coefficients. In dimension 2 the structure of the irreducible ideals–always complete intersections by a classical theorem of Serre–leads by a natural approach to the calculation of Sylvester determinants. We introduce a computer-assisted method (with a minimal intervention by the computer) which succeeds, in degree ≤5, in producing the full sets of equations of the ideals. In the process, it answers affirmatively some questions raised by Cox [Cox, D.A., 2006. Four conjectures: Two for the moving curve ideal and two for the Bezoutian. In: Proceedings of Commutative Algebra and its Interactions with Algebraic Geometry, CIRM, Luminy, France, May 2006 (available in CD media)].