Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS136, part 2: Syzygies and applications to geometry
Time:
Friday, 12/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F-107
30 seats, 56m^2

Presentations
3:00pm - 5:00pm

Syzygies 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)

 

Implicitization of Tensor Product Surfaces via Virtual Projective Resolutions (Part I)

Alexandra Seceleneau
University of Nebraska-Lincoln

In this talk, we address the implicitization problem for tensor product surfaces. A tensor product surface is defined by a parametrization (rational map) $mathbb{P}^1times mathbb{P}^1to mathbb{P}^3$. We consider the problem of computing the defining equation for the tensor product surface based on the polynomials which give its parametrization. Towards this end, we use the residual resultants developed by Busé-Elkadi-Mourrain. Our perspective is informed by the new development of virtual resolutions, which afford the derivation of the implicit equation from a smaller, more manageable algebraic construction than the more standard projective resolutions.

Part I of this talk will discuss the algebraic underpinnings of our method.

 

Implicitization of Tensor Product Surfaces via Virtual Projective Resolutions (Part II)

Eliana Duarte
Otto-von-Guericke Universität Magdeburg

In this talk, we address the implicitization problem for tensor product surfaces. A tensor product surface is defined by a parametrization (rational map) $mathbb{P}^1times mathbb{P}^1to mathbb{P}^3$. We consider the problem of computing the defining equation for the tensor product surface based on the polynomials which give its parametrization. Towards this end, we use the residual resultants developed by Busé-Elkadi-Mourrain. Our perspective is informed by the new development of virtual resolutions, which afford the derivation of the implicit equation from a smaller, more manageable algebraic construction than the more standard projective resolutions.

Part II of this talk will discuss computational considerations and the practical implementation of our method.

 

The Hilbert quasipolynomial of a polynomial ring and generating functions related the Frobenius complexity for various classes of singularities

Florian Enescu
Georgia State University

The talk will present some open questions on the Hilbert quasipolynomial associated to a polynomial ring over a field in finitely many indeterminates, with nonstandard grading. These questions originate in investigations regarding the Frobenius complexity for finitely generated algebras over the integers. The investigation approaches this notion of complexity by analogy to the Hilbert-Samuel and Hilbert-Kunz functions. This is joint work with Yongwei Yao.

 

Generalized Stanley-Reisner rings

Nelly Villamizar
Swansea University

Given a simplicial complex, we study the structure of the subrings of the Stanley-Reisner ring associated to the simplicial complex. These subrings can be seen as the space of spline functions defined on the simplicial complex with higher order continuity conditions accross the faces of the partition. Their ring structure becomes particularly interesting by identifying the ring of continuous splines with the equivariant cohomology ring of a space with a torus action. In the talk, we will explore this identification as well as the the geometric realizations of genealized Stanley-Reisner rings via the description of certain syzygy modules thar encode the smoothness conditions on the splines.