Implicitization of de Jonquières parametrizations

Abstract

One introduces a class of projective parameterizations that resemble generalized de~Jonqui\`eres maps. Any such parametrization defines a birational map $\mathfrak{F}$ of $\pp^n$ onto a hypersurface $V(F)\subset \pp^{n+1}$ with a strong handle to implicitization. From this side, the theory developed here extends recent work of Ben\'{\i}tez and D'Andrea on monoid parameterizations. The paper deals with both the ideal theoretic and effective aspects of the problem. The ring theoretic development gives information on the Castelnuovo-Mumford regularity of the base ideal of $\mathfrak{F}$. From the effective side, we give an explicit formula of $\deg(F)$ involving data from the inverse map of $\mathfrak{F}$ and show how the present parametrization relates to monoid parameterizations.