New Constructions of Cremona Maps

TL;DR

This paper introduces two new methods for constructing Cremona maps, including a duality operation and an iterative process, enabling the creation of explicit families with controlled properties.

Contribution

It presents a characteristic-free duality preserving birationality and an iterative construction for rational maps, expanding tools for Cremona group analysis.

Findings

The Newton complementary dual preserves birationality and acts as an involution on the Cremona group.

An iterative process generates rational maps with controlled topological degree.

Explicit families of Cohen–Macaulay Cremona maps are constructed with specified parameters.

Abstract

One defines two ways of constructing rational maps derived from other rational maps, in a characteristic-free context. The first introduces the Newton complementary dual of a rational map. One main result is that this dual preserves birationality and gives an involutional map of the Cremona group to itself that restricts to the monomial Cremona subgroup and preserves de Jonqui\`eres maps. In the monomial restriction this duality commutes with taking inverse in the group, but is a not a group homomorphism. The second construction is an iterative process to obtain rational maps in increasing dimension. Starting with birational maps, it leads to rational maps whose topological degree is under control. Making use of monoids, the resulting construct is in fact birational if the original map is so. A variation of this idea is considered in order to preserve properties of the base ideal, such as Cohen--Macaulayness. Combining the two methods, one is able to produce explicit infinite families of Cohen--Macaulay Cremona maps with prescribed dimension, codimension and degree.