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

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.Comment: 43 pages, revised version. To appear in Journal of Algebr

On the discriminant scheme of homogeneous polynomials

arXiv reference : arXiv:1210.4697International audienceIn this paper, the discriminant scheme of homogeneous polynomials is studied in two particular cases: the case of a single homogeneous polynomial and the case of a collection of n-1 homogeneous polynomials in n variables. In both situations, a normalized discriminant polynomial is defined over an arbitrary commutative ring of coefficients by means of the resultant theory. An extensive formalism for this discriminant is then developed, including many new properties and computational rules. Finally, it is shown that this discriminant polynomial is faithful to the geometry: it is a defining equation of the discriminant scheme over a general coefficient ring k, typically a domain, if 2 is not equal to 0 in k. The case where 2 is equal to 0 in k is also analyzed in detail

Catégories Dérivées en Cohomologie ℓ\ell-adique

This thesis presents a theory of $\ell$-adic sheaves using the formalism of derived categories.Cette thèse présente une théorie des faisceaux $\ell$-adiques utilisant le formalisme des catégories dérivées