Two Families of Cremona Maps and orthogonal Krall-Jacobi Polynomials

TL;DR

This paper introduces two infinite families of Cremona maps parameterized by a real number, with specific bidegrees, and explores their properties and conjectural existence, supported by computational evidence.

Contribution

It presents new infinite families of Cremona maps with specific bidegrees and investigates their properties and conjectural existence based on computational results.

Findings

Existence of Cremona maps depends on a conjecture for certain dimensions.

Computational evidence suggests the conjecture holds for all n.

Two families of Cremona maps are characterized by their bidegrees.

Abstract

Two infinite families of Cremona maps depending on one real parameter are given. For all integers $n \ge 1$ the first family of Cremona maps consists of group elements in $Bir \left( \mathbb{P}^{n} \right)$ with bidegree $(n, n)$, the second family of Cremona maps consists of group elements in $Bir \left( \mathbb{P}^{2n} \right)$ with bidegree $(2 n, 2 n)$. For the first family and $n \ge 5$, for the second family and $n \ge 3$ the existence of this group elements and the properties depend on a conjecture. But computational results suggest that the conjecture is true for all $n$.