This workshop aims to highlight recent advances in combinatorial algebraic geometry, introduce early career mathematicians to this vibrant field, and foster a collaborative environment as a means to accelerate research discoveries. We will focus on three interconnected topics that have enjoyed groundbreaking advances in the past few years:

• Combinatorial Moduli Spaces

• Combinatorial Hodge Theory

• Toric Degenerations

The workshop will consist of a mixture of survey talks on the current status of the three themes, and more technical research talks. A session of lightning talks to highlight the work of early career participants will complement the programme. Care will be taken not to overpack the schedule, to allow ample interaction between participants.

The workshop will take place in Building 4W Room 1.7 (Wolfson Lecture Theatre).

Tuesday, 2 August 2022

• 09:30–10:15 BST Martin Ulirsch Vector bundles on metric graphs and graph curves

• 10:15 BST Coffee

• 10:45–11:30 BST Joachim Jelisiejew Speculations on the geometry of Hilbert and Quot schemes

• 11:30 BST Lunch

• 13:30–14:15 BST Chris Manon Matroids and the geometry of toric vector bundles

• 14:30–15:15 BST Rohini Ramadas Pullbacks of kappa-classes on M_{0,n}-bar

• 15:15 BST Coffee

• 16:00– BST 5 minute talks

  • Paul Johnson Product formulas for orbifold Hilbert schemes

  • Guilia Iezzi Linear degenerations of Schubert varieties

  • Corin Lee Complexity of cylindrical algebraic decompositions via regular chains

  • Jenna Rajchgot Geometric vertex decomposition of toric ideals of graphs

  • Michael Perlman Calculations involving toric vector bundles

  • Daniele Turchetti Tropical integral models of varieties

  • Gregory Sankaran Blow-ups with mild singularities

  • Flora Poon Geometry of type II locally symmetric spaces

  • Kiumars Kaveh A generalization of the Brianchon–Gram theorem

  • Nolan Schock Intersection theory of tropical compactifications

  • Kuang-Yu Wu Affine subspace concentration conditions

  • Pim Spelier Enriched structures on curves and log geometry

  • Jonathan Lai Polygon mutations and cluster mutations

  • Calla Tschanz Expanded degenerations for Hilbert schemes of pointsExpanded degenerations for Hilbert schemes of point

Thursday, 4 August 2022

• 09:30–10:15 BST Angélica Cueto Splice type surface singularities and their local tropicalizations

• 10:15 BST Coffee

• 10:45–11:30 BST Alex Fink Delta-matroids and the type B permutahedral toric variety

• 11:30 BST Lunch

• 13:30–14:15 BST Dave Jensen Birational geometry of moduli spaces

• 14:30–15:15 BST Adam Gyenge Quot schemes on Kleinian orbifolds and quiver varieties

• 15:15 BST Coffee

• 16:00–16:45 BST Nick Proudfoot Positivity theorems for hyperplane arrangements via intersection theory

Abstracts

Karim Adiprasito (Hebrew University of Jerusalem)

Beyond positivity for lattice polytopes and unimodality of h*

The semigroup algebra of an IDP reflexive lattice polytope was shown to be Gorenstein by Hochster. We compute the fundamental class and conclude a Lefschetz property in arbitrary characteristic. In particular, the h* polynomial has unimodular coefficients. This talk is based on joint work with S. Papadakis, V. Petrotou and J. Steinmeyer.

Alesso Corti (Imperial College London)

Some conjectures and results on toric degenerations of Fano manifolds

Mirror symmetry suggests that certain nice toric degenerations of Fano varieties are in one-to-one correspondence with the torus charts on a “cluster-like” variety. In the talk, I try to make this precise, give some examples, and sketch the surface case where the statement is known (work of my student Wendelin Lutz).

Angélica Cueto (The Ohio State University)

Splice type surface singularities and their local tropicalizations

Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham–Brieskorn–Hamm complete intersections of dimension two. Their construction depends on a weighted graph with no loops called a splice diagram. In this talk, I will report on joint work with Patrick Popescu–Pampu and Dmitry Stepanov (arXiv: 2108.05912) that sheds new light on these singularities via tropical methods, reproving some of Neumann and Wahl's earlier results on these singularities, and showing that splice type surface singularities are Newton non-degenerate in the sense of Khovanskii.

Jyoti Dasgupta (IISER Pune)

Logarithmic connections on toric principal bundles

A connection on a principal bundle generalizes the notion of a directional derivative. We construct the logarithmic Atiyah sequence associated to a principal bundle over a toric variety. We then show the existence of logarithmic connections on toric principal bundles over possibly nonsmooth toric varieties. This is based on ongoing work with Bivas Khan and Mainak Poddar.

Chris Eur (Harvard)

How or when do matroids behave like positive vector bundles?

Motivated by certain toric vector bundles on a toric variety, we introduce "tautological classes of matroids" as a new geometric model for studying matroids. We describe how it unifies, recovers, and extends various results from previous geometric models of matroids. We then explain how it raises several new questions that probe the boundary between combinatorics and algebraic geometry, and discuss how these new questions relate to older questions in matroid theory.

Alex Fink (Queen Mary University of London)

Delta-matroids and the type B permutahedral toric variety

Relationships between matroids and the permutahedral toric variety appear e.g. in matroid Hodge theory. We provide the analogous relationship for delta-matroids, which are Coxeter type B objects. I'll introduce delta-matroids and present some consequences, such as volume polynomials and positivity results for some invariants. This talk is based on work in progress with Chris Eur, Matt Larson and Hunter Spink.

Mark Gross (Cambridge)

Logarithmic and tropical geometry

I will explain the basics of logarithmic geometry, a type of enhancement of scheme theory invented by Illusie-Fontaine and K. Kato in the 1980s as a way of making certain kinds of singular schemes behave as smooth schemes and generalizing the notion of logarithmic differential. I will then explain its connection with tropical geometry via the tropicalization functor from log schemes to polyhedral cone complexes, and try to explain the related notion of Artin fan, an algebraic stack which encodes purely tropical data.

Adam Gyenge (Alfréd Rényi Institute of Mathematics)

Quot schemes on Kleinian orbifolds and quiver varieties

For a finite subgroup Γ ⊂ SL(2, ℂ), we identify fine moduli spaces of certain cornered quiver algebras with orbifold Quot schemes for the Kleinian orbifold ℂ^2/Γ. We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of Γ, taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. This talk is based on joint work with Alastair Craw, Søren Gammelgaard and Balázs Szendrői.

Joachim Jelisiejew (University of Warsaw)

Speculations on the geometry of Hilbert and Quot schemes

The geometry of moduli space of zero-dimensional modules is rather poorly investigated, yet recently became very important for applications in complexity theory, via geometry of tensors and commuting matrices. In the talk I will outline how little is known and present some speculations and open questions. This talk is partially joint work with Klemen Sivic.

Dave Jensen (University of Kentucky)

Birational geometry of moduli spaces

We discuss the birational geometry of various moduli spaces, including moduli of curves, abelian varieties, and Prym varieties. After surveying the current state of research in this area, we will focus on recent work showing that the moduli spaces of curves of genus 22 and 23, and the moduli space of Pryms in genus 13 are of general type. These results use a new perspective on the theory of linear series in tropical geometry to resolve specific cases of the Strong Maximal Rank Conjecture. This talk is based on joint work with Gabi Farkas and Sam Payne.

Eric Katz (The Ohio State University)

The tropical fundamental group

We discuss recent joint work with Kyle Binder on defining the unipotent fundamental group of tropical varieties. This fundamental group arises from the Tannakian formalism using tropical vector bundles with integrable connection. By employing the Orlik–Solomon theorem, we prove that this computes the unipotent completion of the fundamental group of algebraic varieties with smooth tropicalization.

Alex Küronya (Goethe-Universität Frankfurt)

Finite generation of certain valuation semigroups on toric surfaces

The issue of finite generation of (multi)graded rings arising from a geometric context has been a central and notoriously difficult question in the intersection of algebra and geometry. The significant special case of adjoint rings took several decades to solve. Here we discuss valuation algebras associated coming from Newton-Okounkov theory, and consider the special case of toric surfaces. This is an account of joint work with Klaus Altmann, Christian Haase, Karin Schaller, and Lena Walter.

Martina Lanini (Università di Roma Tor Vergata)

Symmetric quivers and symmetric varieties

In this talk, I will report on ongoing joint work with Ryan Kinser and Jenna Rajchgot on varieties of symmetric quiver representations. These varieties are acted upon by a reductive group via change of basis, and it is natural to ask for a parametrisation of the orbits, for the closure inclusion relation among them, for information about the singularities arising in orbit closures. Since the 1980s, the same (and further) questions about representation varieties for type A quivers have been attacked by relating such varieties to Schubert varieties in type A flag varieties (Zelevinsky, Bobinski-Zwara, ...). I will explain that in the symmetric setting it is possible to interpret the above questions in terms of certain symmetric varieties. More precisely, we show that singularities of an orbit closure of a symmetric quiver representation variety are smoothly equivalent to singularities of an appropriate Borel orbit closure in a symmetric variety.

Chris Manon (University of Kentucky)

Matroids and the geometry of toric vector bundles

I'll give an overview of some recent work on the geometry of projectivized toric vector bundles. A toric vector bundle is a vector bundle over a toric variety equipped with an action by the defining torus of the base. As a source of examples, toric vector bundles and their projectivizations provide a rich class of spaces that still manage to admit a combinatorial characterization. I'll begin with a recent classification result which shows that a toric vector bundle can be captured by an arrangement of points on the Bergman fan of a matroid defined by DiRocco, Jabbusch, and Smith in their work on "the parliament of polytopes" of a vector bundle. Then I'll describe how to extract geometric information of the projectivization of the toric vector bundle when this data is nice. I'll discuss the Cox ring, the canonical class, the nef cone, and Fujita's freeness conjectures, focusing on the case when the matroid is uniform. Then I'll describe how these properties interact with natural operations on toric vector bundles. This involves the geometry of the closely related class of toric flag bundles and leads to some combinatorial questions about multilinear operations on matroids. This talk is based on joint work with Kiumars Kaveh, Courtney George, Austin Alderete, and Ayush Tibrewal.

Margarida Melo (Universita Roma Tre)

Tropicalizing universal Jacobians

Moduli spaces of tropical objects can often be obtained as tropicalization of suitable compactifications of classical objects. I’ll show how to realize universal tropical Jacobians on the category of cone stacks both as tropicalization of the non-archimedian universal Jacobians and logarithmic universal Jacobians. This is joint work with S. Molcho, M. Ulirsch, F. Viviani and J. Wise.

Leonid Monin (MPI MiS Leipzig)

Algebraic geometry coming from the theory of oscillators

In studying differential equations, people are especially interested in periodic solutions. One of the methods of approximating a periodic solution of an ordinary differential equation is called harmonic balancing. It amounts to solving a system of polynomial equations. In our work, we study systems of the equations which come from coupled Duffing oscillators and use the theory of Newton Okounkov bodies to find the number of roots of these systems.

Nick Proudfoot (University of Oregon)

Positivity theorems for hyperplane arrangements via intersection theory

I will discuss three recent combinatorial theorems about hyperplane arrangements: the top-heavy conjecture, log concavity of the characteristic polynomial, and non-negativity of the Kazhdan-Lusztig polynomial. Each of these results is proved by studying the cohomology of a projective algebraic variety associated with the arrangement.

Rohini Ramadas (Warwick)

Pullbacks of kappa-classes on M_{0,n}-bar

“Kappa” classes, also known as Miller-Mumford-Morita classes, are tautological cohomology classes on M_{g,n} and on M_{g,n}-bar. They play an important role in the ring-theoretic structure of the cohomology of M_{g,n} and M_{g,n}-bar. For example, in low degree, the cohomology groups of M_{g} are freely generated by monomials in kappa classes.

We consider the subspace K^d of H^{2d}(M_{0,n}-bar) generated by pullbacks of the codimension-d kappa class along all possible forgetful maps. We describe K^d as an S_n-representation, finding a permutation basis. As applications, we find a new permutation basis of the Picard group of M_{0,n}-bar, and in joint work with Rob Silversmith, the first known permutation basis of H^{4}(M_{0,n}-bar).

Rob Silversmith (Warwick)

Cross-ratios and perfect matchings

Given a bipartite graph G (subject to a simple constraint), the "cross-ratio degree” of G is a non-negative integer invariant of G, defined via a simple counting problem in algebraic geometry. I’ll discuss several natural contexts in which cross-ratio degrees arise. I will then present a perhaps-surprising upper bound on cross-ratio degrees in terms of counting perfect matchings. Finally, time permitting, I may discuss the tropical side of the story.

Martin Ulirsch (Goethe–Universität Frankfurt)

Vector bundles on metric graphs and graph curves

Recently the perspective that compact metric graphs are a natural combinatorial, or "tropical", analogue of compact Riemann surfaces has gained significant traction. This is due to its numerous applications in the context of enumerative geometry, the cohomology of moduli spaces, and Brill-Noether theory.

A tropical analogue of line bundles on metric graphs is, by now, well-understood and reflects the various compactifications of the Jacobian over semistable degenerations of compact Riemann surfaces. The goal of this talk is to embark on a journey towards an up-to-now still missing analogue of vector bundles of higher rank on metric graphs. After defining such objects I will talk about a tropical analogue of the Weil–Riemann–Roch–Theorem and of the Narasimhan–Seshadri correspondence. I will also outline a tropicalization procedure that lets us connect this a priori only combinatorial theory with the classical story. As it turns out, this will work best in the case of the Tate curve.

To go beyond the genus one and zero cases, I will then shift gears and propose a new classification of vector bundles on graph curves that simultaneously generalizes the Birkhoff–Grothendieck theorem and Klyachko's classification of toric vector bundles (expanding on its recent reinterpretation as piecewise linear maps to buildings in the work of Kaveh-Manon).

The first half of this talk is based on joint work with Andreas Gross and Dmitry Zakharov.

Bath-LMS symposium website