Titles and Abstracts

Weight properties of resultants and discriminants

Laurent Busé, INRIA Sophia Antipolis

In a book dating back to 1862, Salmon stated a formula giving the first terms of the Taylor expansion of the discriminant of a plane algebraic curve, and from it derived various enumerative quantities for surfaces in the 3-dimensional projective space. In this talk, we will show how this formula can be deduced from a thorough study of weight properties of the discriminant. This leads to introduce the concept of reduced discriminant that extends the reduced resultant introduced by Zariski in 1936.  We will also discuss the deep link there exists with projective enumerative geometry.

Gale duality, blowups and moduli spaces

Ana-Maria Castravet, Université de Versailles

Gale correspondence provides a duality between sets of n points in projective spaces P^s and P^r when n=r+s+2. For small values of s, this duality has a remarkable geometric manifestation: the blowup of P^r at n points can be realized as a moduli space of vector bundles on the blowup of P^s at the Gale dual points. In the spirit of work by Mukai, we explore this realization to describe the birational geometry of blowups of projective spaces at points in very general position.</span

Tropical moduli spaces and the cohomology of M_g

Melody Chan, Brown University

Joint work with Soren Galatius and Sam Payne.  We prove that H^{4g-6}(M_g;Q) grows exponentially with g, where M_g denotes the moduli space of curves of genus g; this disproves conjectures of Kontsevich and Church-Farb-Putman. I will explain our proof, emphasizing our techniques around the combinatorics and computations for boundary complexes.  These techniques are also used in applications to the cohomology of M_{g,n}, in joint work with Faber, Galatius, and Payne, discussed in Payne’s talk on Friday.

A New Approach To Implicitize Rational Surfaces Using Moving Planes and Moving Quadrics

Falai Chen, University of Science and Technology of China

The method of moving surfaces is an effective method to implicitze rational surfaces. However, the theoretic investigation on the validity of the method only succeeds in the case of simple base points with addtional constraints. In this talk, I shall present a new approach to the  validity of implicitization using moving planes and moving quadrics, under the condition that the rational surface contains no base point or base points of local complete intersections. A specific algorithm is also provided to construct the implicitization matrix.

The Birationality of Dual Nef Multiple Mirrors

Patrick Clarke, Drexel University

I will describe the proof of the Batyrev-Nill conjecture on the birationality of mirror families dual to multiple mirror nef-partitions.

Moving Lines, Soccer and Rees Algebras

Carlos D’Andrea, Universitat de Barcelona

In the nineties, several methods for dealing in a more efficient way with the implicitization of rational parametrizations of curves and surfaces were explored in the Computer Aided Geometric Design community. The analysis of these techniques has been a fruitful ground of research in Commutative Algebra and Algebraic Geometry in the last years. In this talk we will review some highlights of this fascinating story which of course has David as one of its main characters.

Tangency and Discriminants

Sandra Di Rocco, KTH

The talk will give a gentle walk through the concept of discriminant of multivariate polynomials and tangential solutions of systems of polynomial equations. The underline  environment and main tool is toric geometry.  The goal is to eventually introduce a  generalized Schäfli method to compute tangential solutions, inspired by the celebrated characterization of hyperdeterminants. The original material is based on joint work with A. Dickenstein and R. Morrison.

Brendan Hassett, ICERM and Brown University

We discuss the rationality of complete intersections of quadric hypersurfaces, focusing on recent results for intersections of two quadrics in five-dimensional space over the real numbers and other non-closed fields.

Stability of Toric Tangent bundles

Milena Hering, University of Edinburgh

In this talk I will give a brief introduction to slope stability and present a combinatorial  criterion for the tangent bundle on a polarised toric variety to be stable in terms of the lattice polytope corresponding to the polarisation. I will then give some applications to toric surfaces and toric varieties of Picard rank 2.  This is joint work with Benjamin Nill and Hendrik Süss.

Jordan Type for Artinian Algebras

Tony Iarrobino, Northeastern University

The Jordan type P_l of an element l in the maximal ideal of an Artinian algebra A is the partition giving the sizes in the Jordan block decomposition of multiplication by l. The Jordan type in general gives more information than whether the pair (A,l) is weak or strong Lefschetz. I will discuss the set of Jordan types possible for a given Artin algebra A and the loci of elements l for which P_l is fixed, as invariants of the algebra A, and discuss their relation with the Hilbert function of A. In particular I will discuss the Jordan type for the free extensions of T. Harima and J. Watanabe — which are generalizations of tensor products. The intent is to overview recent developments, and to pose some open problems.

Reference: arXiv AC/1802.07383 and AC/1806.02767 (v.2 2019) with Pedro Marques and Chris McDaniel.

Computing Singularities of Rational Surfaces

Xiaohong Jia, Chinese Academy of Sciences

We present a symbolic algorithm of computing singularities for a given rational surface using the
technique of moving planes. The algorithm returns
not only the one-dimensional self-intersection loci but also the higher order isolated singularities on
the rational surface. Furthermore, the order of each self-intersection curve or higher-order singularity
is also computed. This algorithm works without the limitation of the number or type of base points
on the rational surface.

Instanton numbers of del Pezzo surfaces

Sheldon Katz, University of Illinois at Urbana-Champaign

In the Mirror Symmetry and Algebraic Geometry book that David and I wrote, we formalized the notion of the instanton numbers n_\beta of a Calabi-Yau threefold X associated to a class \beta in H_2(X,Z), a virtual count of rational curves C \subset X with [C] = \beta.  The total space of the canonical bundle of a del Pezzo surface is Calabi-Yau, and its instanton numbers only depend on the geometry of curves on the original del Pezzo.  In this talk, I give background and prove a general formula for the instanton numbers associated to curve classes with arithmetic genus 0, 1, and 2, using a blend of classical algebraic geometry and modern virtual enumerative geometry.  This talk is based in part on an ongoing project with Jinwon Choi, Michel van Garrel, and Nobuyoshi Takahashi.

Transfinite diameter on varieties

Sione M’au, The University of Auckland

Transfinite diameter is a non-negative constant associated to a compact set, which measures how “separated” its points are.  Transfinite diameter has important applications to potential theory and polynomial approximation. It has a number of other interpretations.  In particular, it is equivalent to the notion of Chebyshev constant, which measures (asymptotically) how small the sup norm of a monic polynomial can get on a compact set.

In several variables, it is straightforward to generalize the definition of transfinite diameter.  But instead of one Chebyshev constant, one has a family of constants parametrized by a simplex – so-called directional Chebyshev constants, where ‘direction’ relates to the relative weights of the different variables in the exponent of the leading monomial of a polynomial. Zaharjuta’s 1975 paper introduced directional Chebyshev constants, and used them to derive an integral formula for transfinite diameter.

David Cox and I defined transfinite diameter on an algebraic subvariety of C^N and generalized Zaharjuta’s theorem to this setting in a 2017 paper.  I will describe what additional tools were needed from computational algebraic geometry. Finally, I will give an application of our results to polynomial interpolation.

Tropical moduli spaces and the top weight Euler characteristic of M_{g,n}

Sam Payne, UT Austin

Joint work with Melody Chan, Carel Faber, and Soren Galatius.  We prove a formula for the generating function (for fixed g and varying n) for the S_n-equivariant Euler characteristic of the top weight cohomology of M_{g,n}, using ideas from graph homology and moduli spaces of tropical curves.  This formula was conjectured by Zagier in 2008, based on examples computed by Faber using the Getzler-Kapranov theory of modular operads. In this talk, I will outline our proof, emphasizing graph-theoretic interpretations of the terms in Zagier’s formula that appear along the way.  I will also highlight some parallels to earlier work of Gorsky and Harer-Zagier on the Euler characteristic of the full cohomology ring of M_{g,n}.

A geometric solution to polynomial equations that arise in an inverse-eigenvalue problem

Tom Sederberg, Brigham Young University

The two most commonly used free-form surface formulations used in CAD and computer graphics are subdivision surfaces (SUBDs) and Non-Uniform Rational B-Splines (NURBS). Numerous formulations have been proposed to combine NURBS and SUBDs into a single representation of which NURBS and SUBDs are each special cases. However, all such attempts experience ugly disfigurations in some cases. This problem is attributed to the eigen structure of refinement matrices. The solution is to define refinement matrices with the desired eigen structure. Algebraically, this inverse-eigenvalue problem reduces to a messy system of polynomial equations. This talk presents a much simpler geometry-based solution.

Smooth Hilbert schemes

Gregory G. Smith, Queen’s University

Consider the Hilbert scheme parametrizing all closed subschemes in projective space with a fixed Hilbert polynomial.  We present numerical conditions on the polynomial that completely characterize when the associated Hilbert scheme is smooth.  In the smooth situation, our explicit description of the subschemes being parametrized also provides new insights into the global geometry.  This talk is based on joint work with Roy Skjelnes.

Triples of lattice polytopes with a given mixed volume

Ivan Soprunov, Cleveland State University

The Bernstein–Khovanskii-Kushnirenko theorem relates the number of isolated solutions to sparse polynomial systems to the mixed volume of the Newton polytopes of the system.

I will talk about an algorithmic approach to classifying triples of lattice polytopes in R^3 with a given value of their mixed volume. Using this approach we produced a complete classification of inclusion-maximal triples of lattice polytopes of mixed volume up to 4. This project is motivated by recent results of Esterov and Gusev on solvability in radicals of generic sparse polynomial systems. This is joint work with Gennadiy Averkov and Christopher Borger.

Lattice Size of Polytopes

Jenya Soprunova, Kent State University

The  lattice size  ls(P) of a lattice polytope P is the smallest  integer n such that  after a unimodular transformation P fits  into the n-dilate of the standard simplex. Earlier work by Schicho, Castryck, and Cools provides an algorithm for computing the lattice size of a polygon based on taking “onion skins” off of P, that is, passing to the convex hull of the interior lattice points of P.  The downside of their algorithm is that in order to find the lattice size, one needs to list all the interior lattice points of P, which is time-consuming.

We explain the connection of the lattice size of P to its successive minima and show that in the case of polygons a reduced lattice basiscomputes the lattice size, which leads to a very fast algorithm for computing  ls(P).

We also provide an algorithm for computing the lattice size of 3-dimensional lattice polytopes and show that for empty lattice tetrahedra there exists a reduced lattice basis for computing the lattice size.

This is joint work with A. Harrison, A. Alajmi, and P. Tierney

Schubert Galois Groups

Frank Sottile, Texas A&M

Problems from enumerative geometry have Galois groups.  Like those from field extensions, these Galois groups reflect the internal structure of the original problem.  The Schubert calculus is a class of problems in enumerative geometry that is very well understood, and may be used as a laboratory to study new phenomena in enumerative geometry.

I will discuss this background, and sketch a picture that is emerging from a sustained study of Schubert problems from the perspective of Galois theory.  This includes a conjecture concerning the possible Schubert Galois groups, a partial solution of the inverse Galois problem, as well as glimpses of the outline of a possible classification of Schubert problems for their Galois groups.

Special Syzygies for Rational Generalized Surfaces of Revolution

Haohao Wang, Southeast Missouri State University

A  surface of revolution with moving axes and angles  is a rational tensor product surface generated from two rational space curves  by rotating one curve (the directrix) around vectors and angles generated by the other curve (the director).   Here we introduce these new kinds of rational generalized surfaces of revolution, provide some interesting examples, and investigate their algebraic and geometric properties.  In particular, we study the base points and syzygies of these rational surfaces. We construct three special syzygies for a surface of revolution with moving axes and angles   from a $\mu$-basis of the directrix, and we show how to compute the implicit equation of these rational surfaces from these three special syzygies. Quaternions and quaternion multiplication are used to represent these surfaces.  Examples are provided to illustrate our theorems and flesh out our algorithms.

Tropical varieties, algorithms, and applications

Josephine Yu, Georgia Tech

I will introduce tropical varieties, algorithms, and applications to computational algebra.  I will discuss connections to elimination theory, Groebner theory, and toric geometry. The emphasis will be on using tropical varieties in numerical algebraic geometry (joint work with Anton Leykin) and higher connectivity of tropical varieties (joint work with Diane Maclagan).