Fibers of rational maps and Rees algebras of their base ideals

TL;DR

This paper investigates the structure of fibers of rational maps between projective spaces and relates them to local cohomology modules of Rees algebras of base ideals, advancing understanding of algebraic geometry and commutative algebra.

Contribution

It introduces a novel connection between the fibers of rational maps and the local cohomology of Rees algebras, providing new insights into their algebraic structure.

Findings

Characterization of fibers via local cohomology modules

Relation between fiber dimensions and Rees algebra properties

New methods for analyzing rational map parameterizations

Abstract

We consider a rational map $\phi: \mathbb{P}_k^{m} \dashrightarrow \mathbb{P}_k^n$ that is a parameterization of an $m$-dimensional variety. Our main goal is to study the $(m-1)$-dimensional fibers of $\phi$ in relation to the $m$-th local cohomology modules of the Rees algebra of its base ideal.