Brill-Noether varieties and set-valued tableaux
by Melody Chan (Brown)
Tuesday, 29 November 2016 at 14:00 in Room 230, Fields Institute
Brill-Noether theory on curves is the classical study of linear series on curves: essentially, maps of curves to projective space. On a smooth compact curve X of genus g, the Brill-Noether variety G^r_d(X) parametrizes linear series on X of rank r and degree d. I will discuss a joint project with Alberto Lopez Martin, Nathan Pflueger, and Montserrat Teixidor i Bigas, in which we use combinatorics related to Buch’s set-valued tableaux, along with Osserman’s machinery of degenerations to Eisenbud-Harris schemes of limit linear series, to give a formula for the genus of G^r_d when it is itself a curve. I’ll also report on recent work related to this result, including new algorithms for set-valued tableaux of skew shape.
Overweight deformations of singular hypersurfaces
by Bernd Schober (Fields)
Tuesday, 22 November 2016 at 14:00 in Room 230, Fields Institute
In my talk I will consider affine hypersurface singularities X defined over an algebraically closed field of characteristic zero. In order to get a deeper understanding of X it is useful to construct an overweight deformation to a better behaved variety. In our setting the latter means one that is given by a binomial prime ideal. By discussing a concrete example, I will illustrate how this can be attained if X is of dimension one. Within this, I will introduce the notion of a weighted characteristic polyhedron. Then I will discuss a generalization for arbitrary dimensional X and state a precise characterization when the construction succeeds to provide an overweight deformation. In the remaining time I will explain the idea for the proof. This is joint work with Hussein Mourtada.
Classes of matrix orbit closures
by Alex Fink (Queen Mary)
Tuesday, 8 November 2016 at 14:00 in Room 230, Fields Institute
There is an obvious, though not canonical, way to record the data of a
hyperplane arrangement in projective space as a matrix of
coefficients. Our protagonist in this talk is the closure of the set
of matrices describing a given arrangement. We discuss formulae for
certain invariants of this variety which depend on the underlying
matroid alone, giving support for the conjecture that its equivariant
Hilbert series has such a formula. This talk includes joint work with
David Speyer and with Andrew Berget.
Chern-Schwarz-Macpherson classes of matroids
by Kristin Shaw (Fields)
Tuesday, 25 October 2016 at 14:00 in Room 230, Fields Institute
Chern-Schwarz-Macpherson classes are one way to extend the notion of Chern classes to singular and non-complete varieties. In this talk, I will provide a combinatorial analogue of these classes for matroids. In this setting, the CSM class is given by Minkowski weights supported on the Bergman fan of the matroid. For matroids arising from hyperplane arrangements over \mathbb{C} these Minkowski weights encode the CSM class of the complement of the arrangement in various compactifications.
One goal in doing this is to obtain a Chow theoretic description of matroid invariants such as the characteristic polynomial, h-vector, and conjecturally Speyer’s g-polynomial and the Tutte polynomial. Secondly, these combinatorial CSM classes can be used to define Chern classes of tropical manifolds which are locally modelled on Bergman fans of matroids.
This is based on joint work with Lucia Lopez de Medrano and Felipe Rincon and also work in progress with Alex Fink and David Speyer.
Waring ranks of homogeneous forms
by Zach Teitler (Boise State)
Tuesday, 18 October 2016 at 14:00 in Room 230, Fields Institute
The Waring rank of a homogeneous form is the number of terms needed to write it as a sum of powers of linear forms. It is related to secant varieties, provides a measure of the complexity of polynomials, and has applications in statistics, sciences, and engineering. I will discuss three topics related to Waring rank. (1) Waring ranks of general forms have been known for some time, but it is also of interest to determine Waring rank of particular forms such as the generic determinant and permanent. I will describe some recent results obtained via algebraic and geometric lower bounds for Waring rank; this is joint work with Jaroslaw Buczynski and with Harm Derksen. (2) A variation of a conjecture of Strassen asserts that the Waring rank of the sum of two forms in independent variables is the sum of the ranks of the summands. I will describe an elementary sufficient condition for a strong version of Strassen’s conjecture. (3) It is an open question to determine the maximum Waring rank occurring among forms of a given degree, in a given number of variables. I will describe an upper bound; this is joint work with Gregoriy Blekherman.
Nef cones of Hilbert schemes of points on surfaces via Bridgeland stability conditions
by Barbara Bolognese (Fields)
Tuesday, 11 October 2016 at 14:00 in Room 230, Fields Institute
Carrying out the Minimal Model Program for moduli spaces is a classical and extremely challenging problem. In this talk, we will deal with a particular moduli space, namely the Hilbert scheme of points on a surface with irregularity zero. After explaining the connection between the birational models of a variety and the combinatorics of its Nef cone, we will show how Bridgeland stability conditions are a powerful machinery to produce extremal rays in the Nef cone of the Hilbert scheme. Time permitting, we will give a complete description of the Nef cone in some examples of low Picard rank. This is joint work with J. Huizenga, Y. Lin, E.Riedl, B. Schmidt, M. Woolf and X. Zhao.
Very, very ample plus nef on toric varieties, or, lattice points in polytopes with long edges
by Christian Haase (FU Berlin)
Tuesday, 27 September 2016 at 14:00 in Room 230, Fields Institute
Let O(D) and O(D’) be an ample and a nef line bundle on a smooth toric variety X, respectively. Oda asked, whether the multiplication map of global sections to sections of O(D+D’) is onto. Annoyingly, this question remains unsolved to this day.
Gubeladze developed a novel convex geometric tool in order to show surjectivity in the case both D and D’ are multiples of the same sufficiently ample divisor. In joint work with Jan Hofmann, we extend Gubeladze’s notion to the general ample+nef case and prove surjectivity if D is really, really ample compared to D’. What we actually do is to add lattice points in lattice polytopes.
Matlis duality on two-dimensional cyclic quotient singularities
by Lars Kastner (Fields)
Tuesday, 20 September 2016 at 14:00 in Room 230, Fields Institute
Ext and Tor modules play important roles in many areas of algebraic geometry, commutative algebra and representation theory. In the toric world, these modules are endowed with a grading by the character lattice. In this talk, we will examine Ext and Tor of two Weil divisors on a two-dimensional cyclic quotient singularity, i.e. an two-dimensional affine toric variety. We will develop a combinatorial description for these modules, and observe a connection of Ext and Tor as Matlis duals.
Algebraic and tropical theta characteristics
by Yoav Len (Fields)
Tuesday, 13 September 2016 at 14:00 in Room 230, Fields Institute
I will discuss the interplay between algebraic and tropical theta characteristics. Various phenomena in algebraic geometry such as tangents to canonical curves, double covers, and Prym varieties are closely related with theta characteristics. By degenerating them in families, we discover analogous constructions in tropical geometry, and links between quadratic forms, covers of graphs and tropical tangents. Time permitting, I will discuss the tropical Prym variety. This is joint work with Dave Jensen.
Loci of hyperelliptic curves in moduli spaces of curves
by Nicola Tarasca (Fields)
Tuesday, 6 September 2016 at 14:00 in Room 230, Fields Institute
In this talk I will discuss some problems in the enumerative geometry of subvarieties of moduli spaces of curves. In joint work with Dawei Chen, we study the extremality of loci of hyperelliptic curves with marked Weierstrass points inside cones of effective classes of high codimension. I will present a closed formula for such classes as result of an ongoing work at the Fields Institute with Renzo Cavalieri.
Resolutions of singularities of Schubert and Richardson varieties
by Laura Escobar (Fields)
Tuesday, 30 August 2016 at 14:00 in Room 230, Fields Institute
In the first part of this talk I will present a combinatorial model describing resolutions of singularities of Schubert varieties, namely tilings by 2-dimensional zonotopes. This is based on joint work with Pechenik, Tenner and Yong. We will then discuss resolutions of singularities for Richardson varieties and talk about the moment polytope of these varieties.
The combinatorics of Artin fans
by Martin Ulirsch (Fields)
Tuesday, 23 August 2016 at 14:00 in Room 1180, Bahen Centre
Artin fans are logarithmic algebraic stacks that are logarithmically étale over an algebraically closed base field. Despite their seemingly abstract definition, the geometry of Artin fans can be described completely in terms of combinatorial objects, so called cone stacks, i.e. geometric stacks over the category of rational polyhedral cones. In this talk, I am going to give an expository account of the theory of Artin fans, focusing on the connection with cone stacks. Given time I will also discuss work-in-progress with R. Cavalieri, M. Chan, and J. Wise, on how cone stacks form a natural framework in order to study the moduli stack of tropical curves.
Combinatorial Algebraic Geometry 