3:00pm - 5:00pmSyzygies and applications to geometry
Chair(s): Laurent Busé (INRIA Sophia Antipolis), Yairon Cid Ruiz (Universitat de Barcelona), Carlos D'Andrea (Universitat de Barcelona)
In this minisymposium, titled "Syzygies and applications to geometry”, we will focus on the striking results and applications that the study of syzygies provides in algebraic geometry, in a wide sense. Topics should include but are not limited to the study of rational and birational maps, singularities, residual intersections and the defining equations of blow-up algebras. We plan to focus on recent progress in this area that result in explicit and effective computations to detect certain geometrical property or invariant. Applications to geometric modeling are very welcome.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Fibers of multi-graded rational maps and orthogonal projection onto rational surfaces
Fatmanur Yildrim
INRIA Sophia Antipolis, France
I will present a new algebraic approach for computing the orthogonal projection of a point onto a rational algebraic surface embedded in the three dimensional projective space, which is a joint work with Nicolás Botbol, Laurent Busé and Marc Chardin. Our approach amounts to turn this problem into the computation of the finite fibers of a generically finite trivariate rational map whose source space is either bi-graded or tri-graded and which has one dimensional base locus: the congruence of normal lines to the rational surface. This latter problem is solved by using certain syzygies associated to this rational map for building matrices that depend linearly in the variables of the three dimensional ambient space. In fact, these matrices have the property that their cokernels at a given point p in three dimensional space are related to the pre-images of the p via the rational map. Thus, they are also related to the orthogonal projections of p onto the rational surface. Then, the orthogonal projections of a point are approximately computed by means of eigenvalues and eigenvectors numerical computations.
Complete intersection points in product of projective spaces
Navid Nemati
Université Pierre et Marie Curie
I will report on ongoing project with Marc Chardin. we study the bigraded Hilbert function of complete intersection sets of points in $mathbb{P}^n times mathbb{P}^m$. We give a sharp lower bound for the stabilization of the bigraded Hilbert function. In addition, we show that, in a specific and pretty large region, the bigraded Hilbert Function only depends upon the degree of the forms defining the points. Finally, we consider the case where the forms defining the points are chosen generically. In this case we show that the natural projections to $mathbb{P}^n$ and $mathbb{P}^m$ are one-to-one.
Fibers of rational maps and Jacobian matrices
Marc Chardin
Université Pierre et Marie Curie
A rational map $varphi$ from a projective space of dimension m to another is defined by homogeneous polynomials of a common degree d. We establish a linear bound in terms of d for the number of (m − 1)-dimensional fibers of $varphi$, by using ideals of minors of the Jacobian matrix. This is joint work with S. Dale Cutkosky and Tran Quang Hoa.
Syzygies and the geometry of rational maps (introductory talk)
Laurent Busé
INRIA Sophia Antipolis
During the past years, the analysis of the syzygies (i.e. the algebraic relations of first order) between the equations defining a geometric object leaded to important advances on many problems lying at the interface of commutative algebra and algebraic geometry, motivated in large part by computer aided assisted computations. In this introductory talk, I will provide an overview of a range of methods and results on the study of the geometric properties of rational maps by means of the syzygies of their defining polynomials, in particular on the understanding of their image and fibers. The computational aspects and the relevance of these results in the field of geometric modeling will also be discussed.