Fibers of rational maps and Jacobian matrices

TL;DR

This paper establishes a linear bound on the number of certain fibers of rational maps using Jacobian matrices, answering an open question and advancing understanding of the geometric properties of these maps.

Contribution

It provides a new linear bound on the number of (m-1)-dimensional fibers of rational maps, linking algebraic properties to geometric fiber structures.

Findings

Bound on the number of fibers is linear in degree d

Uses Jacobian matrix minors to analyze fiber structure

Answers an open question from prior research

Abstract

A rational map $\phi: \mathbb{P}_k^m \dashrightarrow \mathbb{P}_k^n$ is defined by homogeneous polynomials of a common degree $d$. We establish a linear bound in terms of $d$ for the number of $(m-1)$-dimensional fibers of $\phi$, by using ideals of minors of the Jacobian matrix. In particular, we answer affirmatively Question~11 in arXiv:1511.02933v2.