De Jonqui\`eres transformations in arbitrary dimension. An ideal theoretic view

TL;DR

This paper extends de Jonquières transformations from the plane to higher dimensions, focusing on the ideal theoretic properties and homological behavior of the associated base ideals, especially when the Cremona transformation is the identity.

Contribution

It introduces a generalized framework for de Jonquières transformations in arbitrary dimensions with an emphasis on ideal theoretic and homological aspects, particularly for identity-supporting transformations.

Findings

Analysis of the base ideal structure in higher dimensions

Homological properties of the transformation graph when the Cremona map is identity

Use of downgraded sequences of forms as a key tool

Abstract

A generalization of the plane de Jonqui\`eres transformation to arbitrary dimension is studied, with an eye for the ideal theoretic side. In particular, one considers structural properties of the corresponding base ideal and of its defining relations. Useful throughout is the idea of downgraded sequences of forms, a tool considered in many sources for the rounding-up of ideals of defining relations. The emphasis here is on the case where the supporting Cremona transformation of the de Jonqui\`eres transformation is the identity map. In this case we establish aspects of the homological behavior of the graph of the transformation.