## TGiF-Tropical Geometry in Frankfurt

## Next meeting on Zoom

**TGIZ-Tropical geometry in Zoom**

*This is an afternoon seminar series on Tropical Geometry.*

*The next session will be held on the online platform Zoom. Please register by sending an email to one of the organisers by the Jan. 21st. You will then receive the link to the meeting on Jan. 22nd.*

*Those in the SGA mailing list need not to register.*

*Videos of some of the past talks are available via ***this link.**

**this link.**

**January 22nd, 2021 - Second meeting in the Winter Semester 2020/21**

**Schedule **

**14:00-15:00 Alheydis Geiger (Universität Tübingen)**

### 15:00-15:15: Break

**15:15-16:15 Matt Baker (Georgia Institute of Technology)
**

### 16:15-16:30: Break

**16:30-17:30 Daniel Litt (University of Georgia)**

**Details**

* Alheydis Geiger: Deformations of bitangent classes of tropical quartic curves
*

*Abstract:* Over an algebraically closed field a smooth quartic curve has 28 bitangent lines. Plücker proved that over the real numbers we have either 4, 8, 16 or 28 real bitangents to a real quartic curve. A tropical smooth quartic curve has exactly 7 bitangent classes which each lift either 0 or 4 times over the real numbers. The shapes of these bitangent classes have been classified by Markwig and Cueto in 2020, who also determined their real lifting conditions. However, for a fixed unimodular triangulation different choices of coefficients imply different edge lengths of the quartic and these can change the shape of the 7 bitangent classes and might therefore influence their real lifting conditions. In order to prove Plückers Theorem about the number of real bitangents tropically, we have to study these deformations of the bitangent shapes. In a joint work with Marta Panizzut we develope a polymake extension, which computes the tropical bitangents. For this we determine two refinements of the secondary fan: one for which the bitangent shapes in each cone stay constant and one for which the lifting conditions in each cone stay constant. This is still work in progress, but there will be a small software demonstration.

*Matt Baker: Pastures, Polynomials, and Matroids*

*Abstract:* A pasture is, roughly speaking, a field in which addition is allowed to be both multivalued and partially undefined. Pastures are natural objects from the point of view of F_1 geometry and Lorscheid’s theory of ordered blueprints. I will describe a theorem about univariate polynomials over pastures which simultaneously generalizes Descartes’ Rule of Signs and the theory of NewtonPolygons. Conjecturally, there should be a similar picture for several polynomials in several variables generalizing tropical intersection theory. I will also describe a novel approach to the theory of matroid representations which revolves around a canonical universal pasture called the “foundation” that one can attach to any matroid. This is joint work with Oliver Lorscheid.

*Daniel Litt: The tropical section conjecture*

*Abstract:* Grothendieck's section conjecture predicts that for a curve X of genus at least 2 over an arithmetically interesting field (say, a number field or p-adic field), the étale fundamental group of X encodes all the information about rational points on X. In this talk I will formulate a tropical analogue of the section conjecture and explain how to use methods from low-dimensional topology and moduli theory to prove many cases of it. As a byproduct, I'll construct many examples of curves for which the section conjecture is true, in interesting ways. For example, I will explain how to prove the section conjecture for the generic curve, and for the generic curve with a rational divisor class, as well as how to construct curves over p-adic fields which satisfy the section conjecture for geometric reasons. This is joint work with Wanlin Li, Nick Salter, and Padma Srinivasan.

## Future Meetings

**February 19th, 2021 - Third meeting in the Winter Semester 2020/21**

**Schedule **

**14:00-15:00 John Christian Ottem (University of Oslo)**

### 15:00-15:15: Break

**15:15-16:15 Marco Pacini (Universidade Federal Fluminense)**

### 16:15-16:30: Break

**16:30-17:30 Laura Escobar (Washington University in St. Louis)**

**Details**

*John Christian Ottem: Tropical degenerations and stable rationality*

### I will explain how tropical degenerations and birational specialization techniques can be used in rationality problems. In particular, I will apply these techniques to study quartic fivefolds and complete intersections of a quadric and a cubic in P^6. This is joint work with Johannes Nicaise.

*Marco Pacini: A universal tropical Jacobian over the moduli space of tropical curves. *

### Abstract: We introduce polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion-free rank-1 sheaves on a nodal curve. We show how to construct a universal tropical Jacobian by means of polystable divisors on tropical curves. This space can be seen as a tropical counterpart of Caporaso's universal Picard scheme. This is a joint work with Abreu, Andria, and Taboada.

### Laura Escobar: TBA

### Abstract: TBA

**March 12th, 2021 - Fourth meeting in the Winter Semester 2020/21**

**Schedule **

**14:00-15:00 Anthea Monod (Imperial College)**

### 15:00-15:15: Break

**15:15-16:15 Claudia He Yun (Brown University)**

### 16:15-16:30: Break

**16:30-17:30 Daniel Corey (University of Wisconsin)**

**Details**

* TBA.*

## Past online meetings

**4. Dec. 2020 - First meeting in Winter Semester 2020/21**

**Schedule **

**14:00-15:00 Xin Fang (University of Cologne)**

### 15:00-15:15: Break

**15:15-16:15 Man-Wai Cheung (Harvard University)**

### 16:15-16:30: Break

**16:30-17:30 Lara Bossinger (UNAM Oaxaca)**

**Abstracts**

* Xin Fang: Tropical flag varieties - a Lie theoretic approach*

### In this talk I will explain how to use Lie theory to describe the facets of a maximal prime cone in a type A tropical complete flag variety. The face lattice of this cone encodes degeneration structures in Lie algebra, quiver Grassmannians and module categories of quivers. This talk bases on different joint works with (subsets of) G. Cerulli-Irelli, E. Feigin, G. Fourier, M. Gorsky, P. Littelmann, I. Makhlin and M. Reineke, as well as some work in progress.

*Man-Wai Cheung: Polytopes, wall crossings, and cluster varieties*

### Cluster varieties are log Calabi-Yau varieties which are a union of algebraic tori glued by birational "mutation" maps. Partial compactifications of the varieties, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties. However, it is not clear from the definitions how to characterize the polytopes giving compactifications of cluster varieties. We will show how to describe the compactifications easily by broken line convexity. As an application, we will see the non-integral vertex in the Newton Okounkov body of Gr(3,6) comes from broken line convexity. Further, we will also see certain positive polytopes will give us hints about the Batyrev mirror in the cluster setting. The mutations of the polytopes will be related to the almost toric fibration from the symplectic point of view. Finally, we can see how to extend the idea of gluing of tori in Floer theory which then ended up with the Family Floer Mirror for the del Pezzo surfaces of degree 5 and 6. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vianna.

*Lara Bossinger: Tropical geometry of Grassmannians and their cluster structure*

### Abstract: The Grassmannain, or more precisely its homogeneous coordinate ring with respect to the Plücker embedding, was found to be a cluster algebra by Scott in the early years of cluster theory. Since then, this cluster structure was studied from many different perspectives by a number of mathematicians. As the whole subject of cluster algebras broadly speaking divides into two main perspectives, algebraic and geometric, so do the results regarding Grassmannian. Geometrically, the Grassmannian contains two open subschemes that are dual cluster varieties.

### Interestingly, we can find tropical geometry in both directions: from the algebraic point of view, we discover relations between maximal cones in the tropicalization of the defining ideal (what Speyer and Sturmfels call the tropical Grassmannian) and seeds of the cluster algebra. From the geometric point of view, due to work of Fock--Goncharov followed by work of Gross--Hacking--Keel--Kontsevich we know that the scheme theoretic tropical points of the cluster varieties parametrize functions on the Grassmannian.

### In this talk I aim to explain the interaction of tropical geometry with the cluster structure for the Grassmannian from the algebraic and the geometric point of view.

**26th June 2020 - Third meeting in Summer Semester 2020**

**Schedule**

**14:00-15:00: Mark Gross (University of Cambridge)**

*Gluing log Gromov-Witten invariants*

### 15:00-15:15: Break

**15:15-16:15: Luca Battistella (Universität Heidelberg) **

*A smooth compactification of genus two curves in projective space*

### 16:15-16:30: Break

**16:30-17:30: Kalina Mincheva (Yale University)**

*Prime tropical ideals*

**Abstracts**

* Gluing log Gromov-Witten invariants(Mark Gross)*

### I will give a progress report on joint work with Abramovich, Chen and Siebert aiming to understand gluing formulae for log Gromov-Witten invariants, generalizing the Li/Ruan and Jun Li gluing formulas for relative Gromov-Witten invariants.

*A smooth compactification of genus two curves in projective space (Luca Battistella)*

### Questions of enumerative geometry can often be translated into problems of intersection theory on a compact moduli space of curves in projective space. Kontsevich's stable maps work extraordinarily well when the curves are rational, but in higher genus the burden of degenerate contributions is heavily felt, as the moduli space acquires several boundary components. The closure of the locus of maps with smooth source curve is interesting but troublesome, for its functor of points interpretation is most often unclear; on the other hand, after the work of Li--Vakil--Zinger and Ranganathan--Santos-Parker--Wise in genus one, points in the boundary correspond to maps that admit a nice factorisation through some curve with Gorenstein singularities (morally, contracting any higher genus subcurve on which the map is constant). The question becomes how to construct such a universal family of Gorenstein curves. In joint work with F. Carocci, we construct one such family in genus two over a logarithmic modification of the space of admissible covers. I will focus on how tropical geometry determines this logarithmic modification via tropical canonical divisors.

*Prime tropical ideals. (Kalina Mincheva)*

### In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a non-zero prime (tropical) ideal is either empty or consists of a single point.

**29th May 2020 - Second meeting in Summer Semester 2020**

**Schedule**

### 14:00-15:00: Ben Smith (University of Manchester) (cancelled)

*Faces of tropical polyhedra.*

### 15:00-15:15: Break

**15:15-16:15: Yue Ren (University of Swansea) **

*Tropical varieties of neural networks *

### 16:15-16:30: Break

**16:30-17:30: Hannah ****Markwig** (Eberhard Karls Universität Tübingen)

**Markwig**(Eberhard Karls Universität Tübingen)

*The combinatorics and real lifting of tropical bitangents to plane quartics *

**Abstracts**

*Faces of tropical polyhedra (Ben Smith)*

### Tropical polyhedra are tropicalizations of ordinary polyhedra, and have found applications in many areas of pure and applied mathematics. While they have many nice combinatorial properties, the notion of a "face" of a tropical polyhedron has been difficult to define. In this talk, we shall examine the obstacles that arise when considering faces of tropical polyhedra. We also offer a possible solution by defining faces for a special class of tropical polyhedra arising as tropicalisations of blocking polyhedra. We then show how this face structure may be extended to all tropical polyhedra. This is joint work with Georg Loho.

*Tropical varieties of neural networks (Yue Ren)*

In this talk, we introduce tropical varieties arising from neural

networks with piecewise linear activation function. We show how

Stiefel tropical linear spaces correspond to special maxout networks

and compare Speyer's f-Vector Theorem with existing results in machine

learning on their complexity. We briefly touch upon the notion of

Vapnik-Chervonenkis dimension of neural networks and conclude with

some open questions in tropical geometry. This is joint work with

Kathryn Heal (Harvard), Guido Montufar (UCLA + MPI MiS), and Leon Zhang (UC Berkeley).

*The combinatorics and real lifting of tropical bitangents to plane quartics (Hannah Markwig)*

A plane quartic has 28 bitangents. A tropical plane quartic may have

infinitely many bitangents, but there is a natural equivalence relation

for which we obtain precisely 7 bitangent classes. If a tropical quartic

is Trop(V(q)) for a polynomial q in K[x,y] (where K is the field of

complex Puiseux series), it is a natural question where in the 7

bitangent classes the tropicalizations of the 28 bitangents of V(q) are,

or, put differently, which member of the tropical bitangent classes

lifts to a bitangent of V(q), and with what multiplicity. It is not

surprising that each bitangent class has 4 lifts. If q is defined over

the reals, V(q) can have 4, 8, 16 or 28 real bitangents. We show that

each tropical bitangent class has either 0 or 4 real lifts - that is,

either all complex solutions are real, or none. We also discuss further

questions concerning tropical tangents, their combinatorics and their

real lifts. This talk is based on joint work with Yoav Len, and with

Maria Angelica Cueto.

**24th April 2020 - First meeting in Summer Semester 2020**

**Schedule**

**14:00-15:00: Marta Panizzut (Universität Osnabrück)**

*Tropical cubic surfaces and their lines. *

### 15:00-15:15: Break

**15:15-16:15: Jan Draisma (Universität Bern)**

*Catalan-many morphisms to trees-Part I*

### 16:15-16:30: Break

**16:30-17:30: Alejandro Vargas (Universität Bern)**

*Catalan-many morphisms to trees-Part II*

**Abstracts**

*Tropical cubic surfaces and their lines (Marta Panizzut)*

In this talk we investigate different models to study tropical cubic surfaces and their 27 lines.

First we look at smooth tropical cubic surfaces and the combinatorics of their lines in tropical 3-dimensional torus.

We then focus on the tropicalization of the moduli space of del Pezzo surfaces of degree three as in the work of Ren, Shaw and Sturmfels.

Finally we introduce an octanomial model for cubic surfaces. This new normal form is well suited for p-adic geometry,

as it reveals the intrinsic del Pezzo combinatorics of the 27 lines in the tropicalization.

### The talk is based on joint work with Micheal Joswig, Emre Sertöz and Bernd Sturmfels.

*Catalan-many morphisms to trees (Jan Draisma-Alejandro Vargas)*

### Abstract: We report on a several-year project, recently completed, to find a purely combinatorial proof for the result that a genus-g metric graph admits a tropical morphism of genus 1+\lceil g/2 \rceil to a metric tree. The proofs of this result so far have been via specialisation lemmas due to Baker and Caporaso that tropicalize the analogous fact from algebraic geometry.

### We also give a preview on the forthcoming sequel where we count the number of such tropical morphisms in the even genus case and, under a suitable notion of multiplicity, obtain a Catalan number.

### Jan Draisma: introduction to theorem, relation with classical theory

### Alejandro Vargas: key ideas of proof

## Past meetings

**24th January 2020 - Second meeting in Winter Semester 2019/20**

**Schedule**

**13:15-14:15 Karl Christ **(Ben-Gurion University)

*Title: Severi problem and tropical geometry*

### Abstract: The classical Severi problem is to show that the space of reduced and irreducible plane curves of fixed geometric genus and degree is irreducible. In case of characteristic zero, this longstanding problem was settled by Harris in 1986. In the first part of my talk I will give a brief overview of the ideas involved. Then, I will describe a tropical approach to studying degenerations of plane curves, which is the main ingredient to a new proof of irreducibility obtained in collaboration with Xiang He and Ilya Tyomkin. The main feature of the construction is that it works in positive characteristic, where the other known techniques fail.

**14:15-15:00: Coffee Break**

**15:00-16:00**** ****Oliver Lorscheid **(IMPA Rio de Janeiro/MPI Bonn)

*Title: Towards a cohomological understanding of the tropical Riemann Roch theorem*

### Abstract: In this talk, we outline a program of developing a cohomological understanding of the tropical Riemann Roch theorem and discuss the first established steps in detail. In particular, we highlight the role of the tropical hyperfield and explain why ordered blue schemes provide a satisfying framework for tropical scheme theory.

In the last part of the talk, we turn to the notion of matroid bundles, which we hope to be the right tool to set up sheaf cohomology for tropical schemes. This is based on a joint work with Matthew Baker.

**16:15-17:15 Diane Maclagan **(University of Warwick)

*Title: Connectivity of tropical varieties*

### Abstract: The structure theorem for tropical geometry states that the tropicalization of an irreducible subvariety of the algebraic torus over an algebraically closed field is the support of a pure polyhedral complex that is connected through codimension one. This means that the hypergraph whose vertices correspond to facets of the complex, and whose hyperedges correspond to the ridges, is connected. In this talk I will discuss joint work with Josephine Yu showing that this hypergraph is in fact d-connected (when the complex has no lineality space). This can be thought of as a generalization of Balinski's theorem on the d-connectivity of the edge graph of a d-polytope. A key ingredient of the proof is a toric Bertini theorem of Fuchs, Mantova, and Zannier, plus additions of Amoroso and Sombra.

**31st October 2019 - First meeting in Winter Semester 2019/20**

**Schedule**

**14:00-15:00 Enrica Mazzon **(Max-Planck-Institute Bonn)

*Title: *Tropical affine manifolds in mirror symmetry

### Abstract: Mirror symmetry is a fast-moving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, it suggests that certain geometrical objects (complex Calabi-Yau manifolds) should come in pairs, in the sense that each of them has a mirror partner and the two share interesting geometrical properties. In this talk I will introduce some notions relating mirror symmetry to tropical geometry, inspired by the work of Kontsevich-Soibelman and Gross-Siebert. In particular, I will focus on the construction of a so-called “tropical affine manifold” using methods of non-archimedean geometry, and the guiding example will be the case of K3 surfaces and some hyper-Kähler varieties. This is based on a joint work with Morgan Brown and a work in progress with Léonard Pille-Schneider.

**15:00-15:30: Coffee Break**

**15:30-16:30**** ****Christoph Goldner **(Tübingen)

*Title: Tropical mirror symmetry for ExP*^{^1}

^{^1}