On the closed image of a rational map and the implicitization problem

TL;DR

This paper studies the image of a rational map in algebraic geometry, providing formulas for its degree, relating its defining ideal to algebraic structures, and deriving implicit equations under specific conditions.

Contribution

It introduces new methods to compute the degree of the image, links the defining ideal to symmetric and Rees algebras, and derives implicit equations for hypersurfaces with certain base point conditions.

Findings

Degree formula for generically finite maps with isolated base points

Relations between defining ideal and algebraic structures

Implicit equations for hypersurfaces with specific base points

Abstract

In this paper, we investigate some topics around the closed image $S$ of a rational map $\lambda$ given by some homogeneous elements $f_1,...,f_n$ 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 $f_1,...,f_n$ 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 $I=(f_1,...,f_n) \subset A$, 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.