On the homology of two-dimensional elimination

TL;DR

This paper investigates the homology of two-dimensional elimination by analyzing birational maps defined by almost complete intersection ideals, providing new methods for calculating Chern numbers and explicit equations for base ideals.

Contribution

It introduces an effective method to compute Chern numbers using Hilbert coefficients and determines the defining equations of base ideals for degrees up to 5.

Findings

Birationality characterized by equality of two Chern numbers.

Provided a computational approach for Sylvester determinants.

Full set of defining equations for base ideals of degree ≤ 5.

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 of their calculation in terms of certain Hilbert coefficients. In dimension two the structure of the irreducible ideals leads naturally to the calculation of Sylvester determinants via a computer-assisted method. For degree at most 5 we produce the full set of defining equations of the base ideal. The results answer affirmatively some questions raised by D. Cox.