On Rees algebras and de Jonqui\`eres transformations

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.