Computational Algebra, Algebraic Geometry and Applications

Abstracts of Talks

- On a special family of Fano varieties, Carolina Araujo
The geometry of a quartic del Pezzo surface S is very well understood. Embedded as a smooth complete intersection of two quadric hypersurfaces in ℙ

^{4}, the surface S contains exactly 16 lines. It can also be described as the blowup of ℙ^{2}at 5 points in general linear position. In fact, there are 16 different ways to realize S as such blowup: for each line ℓ ⊂ S, there is one such blowup under which ℓ is the transform of the unique conic through the blown up points.In this talk, we will explain how this picture generalizes to arbitrary even dimension. Given an even positive integer n = 2m, we consider the variety G parametrizing (m - 1)-planes in a smooth complete intersection of two quadrics in ℙ

^{2m+2}. This is a Fano variety of dimension n that can also be described as a small modification of the blowup of ℙ^{n}at n + 3 points in general linear position. We show that there are 2^{n+2}different ways to realize G in this manner, one for each of the 2^{n+2}distinct m-planes contained in the complete intersection of two quadrics, and describe these birational maps explicitly. This is a joint work with Cinzia Casagrande. - Discriminant of complete intersections in the 3-dimensional
projective space, Laurent Busé
In this talk, the definition and the formalism of the discriminant of complete intersections in the three-dimensional projective space will be discussed. We will provide a definition of the discriminant by means of the theory of resultants. This approach allows to work in a large class of coefficient rings and to obtain formal properties and computational rules from those of resultants. This talk reports on joint works with Jean-Pierre Jouanolou (University of Strasbourg) and Ibrahim Nonkane (University of Ouagadogou, Burkina Faso).

- Multidimensional Residues and Applications, Eduardo Cattani
I will give a brief, and necessarily incomplete, survey of Alicia Dickenstein’s many contributions to the study of multidimensional residues and their applications to elimination theory and generalized hypergeometric functions.

- Bigraded Commutative Algebra and Implicitization, David Cox
This talk will survey the algebra and geometry of bigraded polynomials in four variables. The algebra involves free resolutions and approximation complexes, while the geometry is primarily concerned with surfaces in projective 3-space, with the goal of finding an implicit equation or matrix representation. I will mention works by various authors, including Cox, Goldman, Zhang, Galligo, Elkadi, Le, Botbol, Dohm, Schenck, Seceleanu, Validashti, Duarte and of course Dickenstein.

- On the stability of polynomial logarithmic differential one-forms,
Fernando Cukierman
A result due to Omegar Calvo states that the polynomial logarithmic one-forms of a certain multi-degree constitute an irreducible component of the scheme of integrable one-forms on a projective space. I plan to talk about a new proof of this theorem, obtained with Javier Gargiulo and Cesar Massri. Our method is based on tangent space calculations and also implies that the logarithmic irreducible components are generically reduced.

- Visualising toric vector bundles, Sandra di Rocco
Klaychko’s characterization of toric vector bundles intrinsically carries the definition of associated polytopes, extending the corresponding theory of line bundles. Positivity properties, as global generation and generation of jets, are visualised in terms of convex properties of the corresponding polytopes and provide useful criteria. I will illustrate this correspondence and show how these criteria lead to proving and disproving connections between various notions of positivity and cohomology vanishing. This is joint work with G. Smith and K. Jabbusch.

- On the combinatorics and computation of sparse resultants,
Gabriela Jeronimo
Recently, a new definition for the sparse resultant associated to n+1 Laurent polynomials in n variables with given supports has been introduced. It produces more uniform statements and enables the extension of previous results to arbitrary collections of supports. We will discuss factorization formulas for initial forms and for degrees of homogenization of the new sparse resultant, and we will present a simplified recursive formulation to produce Macaulay stlye formulae for this resultant. This is a joint work with Carlos D’Andrea and Martín Sombra.

- Algebraic models in systems biology, Reinhard Laubenbacher
As biology has become a data-rich science, more biological phenomena have become amenable to modeling and analysis using mathematical and statistical methods. At the same time, more mathematical areas have developed applications in the biosciences, in particular algebra, discrete mathematics, topology, and geometry. This talk will present some case studies from algebra and discrete mathematics applied to the construction and analysis of dynamic models of biological networks. Some emerging themes will be highlighted, outlining a broader research agenda at the interface of biology and algebra and discrete mathematics.

- Binomial ideals, Laura Matusevich
A binomial is a polynomial (in several commuting variables) with at most two terms. Binomial ideals, ideals generated by binomials, turn out to have a very rich combinatorial structure. I will illustrate this by showing different ways of decomposing a binomial ideal as an intersection of (simpler) binomial ideals. How we define “simpler” is very important: the more combinatorial requirements we ask for, the more challenging the computation becomes. The surprising fact that meaningful and effectively computable binomial decompositions exist is due to Eisenbud and Sturmfels. I will recall their results, and then survey the combinatorial progress that has occurred in the last ten years.

- Isospectral lens spaces, Ehrhart’s theory and associated toric
varieties, Roberto Miatello
We will discuss results on isospectrality of lens spaces (joint with Emilio Lauret and Juan Pablo Rossetti) from the point of view of Ehrhart’s theory, together with implications involving the associated toric varieties.

- From residues to poles, Bernard Mourrain
Residues are linear forms, which encode the algebraic properties of a complete intersection or an Artinian Gorenstein algebra. In several problems, we are interested in computing the poles of the residue or the roots of the associated Artinian Gorenstein Algebra from a partial knowledge of its residue. In this talk, we will recall the algebraic properties of residues and present a fast algorithm for the computation of the poles of a residue and for sparse decomposition of series. We will illustrate the method by examples related to sparse interpolation, linear recurrence relations, quadrature formulas,...

- Computing the topology of real algebraic curves, Marie-Françoise
Roy
Let P ∈ ℤ[X,Y ] be a square-free polynomial of total degree d and bitsize of coefficients τ. Our main result is a deterministic algorithm for the computation of the topology of the curve defined by P (i.e a straight-line planar graph isotopic to the curve defined by P inside ℝ

^{2}) in Õ(d^{5}τ + d^{6}) bit operations where Õ means that we ignore logarithmic factors in d and τ. Our method avoids entirely a generic shear (joint work with Daouda Niang Diatta, Seny Diatta, Fabrice Rouillier and Michael Sagraloff). - Elimination and sparse systems, Juan Sabia
In this talk, an algorithmic approach to compute projections of algebraic varieties defined by sparse generic systems will be discussed. The results shown here are part of joint works with María Isabel Herrero and Gabriela Jerónimo.

- Birational maps with Cohen–Macaulay graphs, Aron Simis
A rational map whose source and image are projectively embedded varieties has a Cohen–Macaulay graph if the Rees algebra of one (hence any) of its base ideals is a Cohen–Macaulay ring. If the map is birational onto the image one considers how this property forces an upper bound on the degree of a representative of the map. In the plane case a complete description is given of the Cremona maps with Cohen–Macaulay graph, while in arbitrary dimension n it is shown that a Cremona map with Cohen–Macaulay graph has degree at most n

^{2}. - A-Discriminant Coamoebas in Dimension Three, Frank Sottile
The coamoeba of a subvariety V of the torus is the set of arguments of points of V. The closure of a coamoeba is the union of coamoebae of its initial schemes, which are indexed by the cones in its tropical variety. We describe the closure of the coamoeba of a reduced A-discriminant in dimension three. Surprisingly, only the initial schemes coming from elements of the Gale dual of A contribute, which leads to an inductive construction of an A-discriminat as an overlapping union of polyhedra. This is joint work with Mounir Nisse.

- Cameras and their Distortions, Bernd Sturmfels.
We discuss two current projects at the interface of computer vision and algebraic geometry. Work with Joe Kileel, Zuzana Kukelov and Tomas Pajdla introduces the distortion varieties of a given projective variety. These are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions, the case of most interest for modeling cameras with image distortion. Work with Jean Ponce and Matthew Trager develops multi-view geometry for algebraic cameras that are represented by congruences in the Grassmannian of lines in 3-space.