The resultant divisor is negative

TL;DR

This paper investigates the birational geometry of a parameter space for polynomial pairs, demonstrating that the minimal model program contracts the resultant divisor, revealing its geometric significance.

Contribution

It establishes that the MMP can be applied to this space and that it contracts the resultant divisor, providing new insights into its geometric role.

Findings

The MMP can be run on the parameter space.

The resultant divisor is contracted during the MMP.

The space's birational geometry is characterized by this contraction.

Abstract

Fix two integers $1\leq d<e$. We study the birational geometry of a parameter space for pairs of homogeneous polynomials of degrees $d$ and $e$ in two variables (in which the higher degree polynomial is well defined only up to a multiple of the lower degree polynomial). We show that one can run the MMP on this space, and that it eventually contracts the resultant divisor.