ON THE CLOSED IMAGE OF A RATIONAL MAP AND THE IMPLICITIZATION PROBLEM

By Laurent Busé, J. Jouanolou. 2002

99 citations recorded by the source corpus.

In this paper, we investigate some topics around the closed image $S$ of a rational map $\lambda$ given by some homogeneous elements $f_1,\ldots,f_n$ of the same degree in a graded algebra $A$. We first compute the degree of this closed image in case $\lambda$ is generically finite and $f_1,\ldots,f_n$ define isolated base points in $\Proj(A)$. We then relate the definition ideal of $S$ to the symmetric and the Rees algebras of the ideal $I=(f_1,\ldots,f_n) \subset A$, and prove some new acyclicity criteria for the associated approximation complexes. Finally, we use these results to obtain the implicit equation of $S$ in case $S$ is a hypersurface, $\Proj(A)=\PP^{n-2}_k$ with $k$ a field, and base points are either absent or local complete intersection isolated points.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Oncology

View Paper: https://www.semanticscholar.org/paper/552ceafda985657d386d386724e5c442c2d476bd

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In Algebra and Number Theory

1. Andrew R. Kustin
2. C. Polini
3. Bernd Ulrich

In Computational Theory and Mathematics

1. Andrew R. Kustin
2. C. Polini
3. Bernd Ulrich

Citations in the corpus

• Rational normal scrolls and the defining equations of Rees algebras (2008, background): “The computation of the defining equations of the fiber ring F is called the implicitization problem for the curve C; see, for example, [3,5,6,13].”
• Syzygies and the Rees algebra (2008, unknown): “5 of [4].”