The implicit equation of a multigraded hypersurface

In this article we analyze the implicitization problem of the image of a rational map $\phi: X --> P^n$, with $T$ a toric variety of dimension $n-1$ defined by its Cox ring $R$. Let $I:=(f_0,...,f_n)$ be $n+1$ homogeneous elements of $R$. We blow-up the base locus of $\phi$, $V(I)$, and we approximate the Rees algebra $Rees_R(I)$ of this blow-up by the symmetric algebra $Sym_R(I)$. We provide under suitable assumptions, resolutions $\Z.$ for $Sym_R(I)$ graded by the torus-invariant divisor group of $X$, $Cl(X)$, such that the determinant of a graded strand, $\det((\Z.)_\mu)$, gives a multiple of the implicit equation, for suitable $\mu\in Cl(X)$. Indeed, we compute a region in $Cl(X)$ which depends on the regularity of $Sym_R(I)$ where to choose $\mu$. We also give a geometrical interpretation of the possible other factors appearing in $\det((\Z.)_\mu)$. A very detailed description is given when $X$ is a multiprojective space.Comment: 19 pages, 2 figures. To appear in Journal of Algebr

Compactifications of rational maps, and the implicit equations of their images

In this paper we give different compactifications for the domain and the codomain of an affine rational map $f$ which parametrizes a hypersurface. We show that the closure of the image of this map (with possibly some other extra hypersurfaces) can be represented by a matrix of linear syzygies. We compactify $\Bbb {A}^{n-1}$ into an $(n-1)$-dimensional projective arithmetically Cohen-Macaulay subscheme of some $\Bbb {P}^N$. One particular interesting compactification of $\Bbb {A}^{n-1}$ is the toric variety associated to the Newton polytope of the polynomials defining $f$. We consider two different compactifications for the codomain of $f$: $\Bbb {P}^n$ and $(\Bbb {P}^1)^n$. In both cases we give sufficient conditions, in terms of the nature of the base locus of the map, for getting a matrix representation of its closed image, without involving extra hypersurfaces. This constitutes a direct generalization of the corresponding results established in [BuseJouanolou03], [BuseChardinJouanolou06], [BuseDohm07], [BotbolDickensteinDohm09] and [Botbol09].Comment: 2 images, 28 pages. To appear in Journal of Pure and Applied Algebr

Implicitization of rational maps

Motivated by the interest in computing explicit formulas for resultants and discriminants initiated by B\'ezout, Cayley and Sylvester in the eighteenth and nineteenth centuries, and emphasized in the latest years due to the increase of computing power, we focus on the implicitization of hypersurfaces in several contexts. Our approach is based on the use of linear syzygies by means of approximation complexes, following [Bus\'e Jouanolou 03], where they develop the theory for a rational map $f:P^{n-1}\dashrightarrow P^n$. Approximation complexes were first introduced by Herzog, Simis and Vasconcelos in [Herzog Simis Vasconcelos 82] almost 30 years ago. The main obstruction for this approximation complex-based method comes from the bad behavior of the base locus of $f$. Thus, it is natural to try different compatifications of $A^{n-1}$, that are better suited to the map $f$, in order to avoid unwanted base points. With this purpose, in this thesis we study toric compactifications $T$ for $A^{n-1}$. We provide resolutions $Z.$ for $Sym_I(A)$, such that $\det((Z.)_\nu)$ gives a multiple of the implicit equation, for a graded strand $\nu\gg 0$. Precisely, we give specific bounds $\nu$ on all these settings which depend on the regularity of \SIA. Starting from the homogeneous structure of the Cox ring of a toric variety, graded by the divisor class group of $T$, we give a general definition of Castelnuovo-Mumford regularity for a polynomial ring $R$ over a commutative ring $k$, graded by a finitely generated abelian group $G$, in terms of the support of some local cohomology modules. As in the standard case, for a $G$-graded $R$-module $M$ and an homogeneous ideal $B$ of $R$, we relate the support of $H_B^i(M)$ with the support of $Tor_j^R(M,k)$.Comment: PhD. Thesis of the author, from Universit\'e de Paris VI and Univesidad de Buenos Aires. Advisors: Marc Chardin and Alicia Dickenstein. Defended the 29th september 2010. 163 pages 15 figure

MS

thesisThe gilsonite and ozokerite deposits of the Unita Basin, Utah, were investigated geochemically. Soil samples were collected along traverses across productive veins in several areas. These soil samples were analysed for gilsonite and ozokerite. The gilsonite and ozokerite content of the soil samples was related to vein proximity. The veins investigated were the Cowboy vein, the Chepetta lode, the Carbon lode, the Rainbow vein, the Pariette vein, and the Soldier Summit ozokerite deposit. Three laboratory procedures were developed to separate gilsonite from the soil samples. All were based on the difference in specific gravity between gilsonite, which has a specific gravity of 1.04, and soil fraction, which has a specific gravity of about 2.0 - 2.6. In one procedure tetrabromoethane in benzene was the heavy liquid medium. In the other procedure carbon tetrachloride was the heavy liquid medium. In each of these procedures, the gilsonite was floated. In the third procedure, the gilsonite was separated from the soil by panning. In each procedure the gilsonite fraction was weighed, and the gilsonite content of each sample was expressed as parts gilsonite per million parts of sample by weight. The precision of the procedures was determined by replicate analyses. The standard deviation of the derived result in the calculation for PPM gilsonite is +-16.6%. Geochemical anomalies were recognized in each instance. Background values ranged from 200 PPM to 1,000 PPM. Anomalies ranged from 2,910 PPM to 264,000 PPM, and the contrast raged from 16 t0 545. In addition to geochemical anomalies disclosed over each vein, several anomalies at a distance from known veins were observed. These anomalies may reflect covered veins. Gilsonite content and dispersion halos surrounding the veins are due to weathering and erosion of gilsonite veins

Maladie de Lyme, syndrome autistique et traitement antibiotique : une réflexion à partir d'un cas

International audienceFollowing an overview of the definition of autism and Lyme disease, a clinical case will be presented of a young subject (aged 14 years and a half) diagnosed with autism, and characterized by a typical development in the first year of life, followed by a general deterioration of abilities shortly after diagnosis of the onset of Lyme disease. The subject presented positive reactivity for Mycoplasma Pneumoniae. According to recent studies, misdiagnosis of initial symptoms of Lyme disease and delayed treatment can lead to persistent interactive coinfections in the organism which may result in chronic pathologies. Thus, a link could exist between a chronic infectionwith Mycoplasma Pneumoniae and an atypical symptomology similar to autism. So the hypothesis proposed is that there may be a correlation between Lyme disease and autism and that long-term antibiotic therapy may be an effective treatment. Following a clinical presentation of the history of this young patient, the evolution of the subject's symptomology and behavior during one year of antibiotic therapy will be described.A la suite d'un rappel de la définition de l'autisme infantile et de la maladie de Lyme, un cas clinique sera développé. Il s'agit d'un jeune patient avec autisme âgé de 14 ans et demi, caractérisé par un développement initial typique pendant sa première année puis par une dégradation générale de ses aptitudes peu après avoir été atteint de la maladie de Lyme. A la suite d'une présentation sous la forme d'une vignette clinique, nous aborderons l'évolution de la symptomatologie et du comportement de ce jeune pendant un an d'antibiothérapie. Cette présentation de cas permettra finalement d'interroger la définition de l'autisme, son étiologie et la validité du diagnostic établi initialement pour ce patient