Degree of Ball Maps with Maximum Geometric Rank

TL;DR

This paper establishes a degree bound for rational proper maps between complex balls with maximum geometric rank, showing such maps cannot exceed degree n+1 under specific dimension and rank conditions.

Contribution

It provides a new upper bound on the degree of rational proper maps between balls with maximum geometric rank, advancing understanding of their geometric properties.

Findings

Degree of maps with maximum geometric rank is bounded by n+1.

Proper maps with specified dimensions and rank cannot have degree higher than this bound.

Results apply to rational proper maps between complex balls.

Abstract

This work focuses on the degree bound of maps between balls with maximum geometric rank and minimum target dimension where this geometric rank occurs. Specifically, we show that rational proper maps between $\mathbb{B}_n$ and $\mathbb{B}_N$ with $n \geq 2$, $N = \frac{n(n+1)}{2}$, and geometric rank $n-1$ cannot have a degree of more than $n+1$.