# 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.

## Summary (1 min read)

### 1. Introduction and notation.

- These have been variously studied by several authors, some listed in the references.
- Ours is a modification of this method, hereby called birational downgrading, by which the authors use the forms defining the inverse map rather than the usual procedures in the literature.
- The main result yields a set of Rees equations candidates for a set of minimal generators, generating an ideal having as a minimal prime component the entire Rees ideal.

### 3.3. The inclusion case.

- This triggers a natural injection R(J) ⊂ R(I) of Rees algebras.
- Thus, in principle, this would give information about the defining Rees equations of J out of these of the base ideal I.
- Setting up explicit presentations requires moving around variables, so the ultimate computational advantage is not so clear.
- One may ask how implicitization may profit from this simple situation of linkage in a coarse sense.

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