Matrix representations for toric parametrizations

In this paper we show that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over P^2 established in [BJ03] and [BC05]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case T = P^1 x P^1 in detail and give numerous examples.Comment: 20 page

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

Intersection entre courbes et surfaces rationnelles au moyen des représentations implicites matricielles

National audienceDans cet article, on introduit une nouvelle représentation implicite des courbes et des surfaces paramétrées rationelles, représentation qui consiste pour l'essentiel à les caractériser par la chute de rang d'une matrice plutôt que par l'annulation simultanée d'une ou plusieurs équations polynomiales. On montre comment ces représentations implicites, que l'on qualifiera de matricielles, établissent un pont entre la géométrie et l'algèbre linéaire, pont qui permet de livrer des problèmes géométriques à des algorithmes classiques et éprouvés d'algèbre linéaire, ouvrant ainsi la possibilité d'un traitement numérique plus robuste. La contribution de cette approche est discutée et illustrée sur des problèmes importants de la modélisation géométrique tels que la localisation (appartenance d'un point à un objet), le calcul d'intersection de deux objets, ou bien encore la détection d'un lieu singulier

Syzygies and singularities of tensor product surfaces of bidegree (2,1)

Let U be a basepoint free four-dimensional subspace of the space of sections of O(2,1) on P^1 x P^1. The sections corresponding to U determine a regular map p_U: P^1 x P^1 --> P^3. We study the associated bigraded ideal I_U in k[s,t;u,v] from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation of the image p_U(P^1 x P^1), via work of Buse-Jouanolou, Buse-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex. In four of the six cases I_U has a linear first syzygy; remarkably from this we obtain all differentials in the minimal free resolution. In particular this allows us to describe the implicit equation and singular locus of the image.Comment: 35 pages 1 figur

The surface/surface intersection problem by means of matrix based representations

International audienceEvaluating the intersection of two rational parameterized algebraic surfaces is an important problem in solid modeling. In this paper, we make use of some generalized matrix based representations of parameterized surfaces in order to represent the intersection curve of two such surfaces as the zero set of a matrix determinant. As a consequence, we extend to a dramatically larger class of rational parameterized surfaces, the applicability of a general approach to the surface/surface intersection problem due to J.~Canny and D.~Manocha. In this way, we obtain compact and efficient representations of intersection curves allowing to reduce some geometric operations on such curves to matrix operations using results from linear algebra

Toric Geometry

Toric Geometry plays a major role where a wide variety of mathematical fields intersect, such as algebraic and symplectic geometry, algebraic groups, and combinatorics. The main feature of this workshop was to bring people from these area together to learn about mutual, possibly up till now unnoticed similarities in their respective research

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