On Rees algebras and de Jonqui\`eres transformations

TL;DR

This paper explores higher-dimensional analogs of de Jonquières transformations, focusing on implicitization of their Rees algebras and addressing a conjecture related to their Rees rings.

Contribution

It provides a method to implicitize the Rees algebra of higher-dimensional de Jonquières maps and confirms a conjecture about their Rees rings.

Findings

Implicitization of Rees algebras for higher-dimensional de Jonquières maps.

Extension of results to generalized de Jonquières transformations.

Resolution of a conjecture by Ramos and Simis.

Abstract

We recall a higher dimension analog of the classic plane de Jonqui\`eres transformations, as given by Hassanzadeh and Simis. Such a parameterization defines a birational map from $\mathbb{P}^{n-1}$ to a hypersurface in $\mathbb{P}^{n}$, and a natural question that arises is how to obtain its implicit equation. We pass from the image of this map to its graph, and implicitize the Rees algebra of the ideal of the de Jonqui\`eres map when its underlying Cremona support is tame. We then consider the Rees rings of ideals of generalized de Jonqui\`eres transformations, and answer a conjecture of Ramos and Simis.