# Workshop on Tropical varieties and amoebas in higher dimension

April 16 - 20, 2018

Venue: KTH Royal Institute of Technology, the KTH library, Osquars Backe 31:

For Monday and Friday, the room is Salongen.

For Tuesday-Thursday, the room is Sydöstra Galleriet.

This workshop is the final conference of the Spring programme "Tropical Geometry, Amoebas and Polytopes", and is intended to cover a broad range of topics that will be of interest during the programme, in particular tropical algebra and applications; combinatorics, polytopes, and complexity; moduli spaces of curves and mirror symmetry; and tropical geometry and amoebas in higher dimension. The conference is sponsored by the Simons Foundation.

Participation in the conference is open to everyone (and in particular not restricted to participants in the programme).

For hotel booking, travel questions or any other administrative matters please contact Ann-Britt Isaksson Öhman: annbritt@kth.se.

## Preliminary week schedule:

### Monday

8:50 Welcome and Registration

9-9:50 Frank Sottile

Title: Describing Amoebas

Abstract: Amoebas and coamoebas are curvilinear cousins of tropical varieties, which consist of the lengths and the arguments, respectively, of an algebraic subvariety of the torus. Purbhoo's Nullstellensatz shows that the amoeba of a variety is the intersection of amoebas of hypersurfaces coming from its ideal. Subsequent work has involved approximating amoebas and showing that zero-dimensional amoebas are the finite intersection of hypersurface amoebae.

Starting from the observation that amoebas and coamoebas are semi-algebraic sets, I propose the problem of finding a description of these tropical cousins as semi-algebraic sets. This appears hard, even for curves in low-dimensional tori. I will also discuss progress with Nisse towards classifying those amoebas that are the finite intersection of hypersurface amoebas.

10-10:50 August Tsikh

Title: On the structure and amoebas of discriminants

11-11:30 coffee

11:30-12:20 Johannes Nicaise

Title: A motivic Fubini theorem for the tropicalization map

Abstract: This talk is based on joint work with Sam Payne (arXiv:1703.10228). Kontsevich and Soibelman's motivic upgrade of Donaldson-Thomas theory produces refined curve counting invariants by means of motivic vanishing cycles of potential functions. In order to get a coherent theory, Kontsevich-Soibelman and Davison-Meinhardt have conjectured formulas for the motivic vanishing cycles of special types of functions. I will explain how one can deduce these formulas from a combination of Hrushovski-Kazhdan motivic integration and tropical geometry. The talk will be less intimidating than the abstract.

12:30-14:00 lunch

14:00-14:20 Yue Ren

Title: Tropical basis verification

Abstract: We will briefly discuss some quick countercertificates for tropical bases, i.e. tests which if positive imply that a given generating set is no tropical basis, relying on projections and (stable) intersections of tropical varieties. We will apply them on a few open problems in the literature and discuss their ramifications in cases for which they fail.

This is ongoing work with Paul Goerlach and Jeff Sommars.

14:30-14:50 Lionel Lang

Title: An analogue of Hilbert's sixteen problem for amoebas

Abstract: Consider the space of curves C defined by Laurent polynomials with a fixed Newton polygon. We address the following problem: classify the topological pairs (C,cr(C)) where cr(C) is the set of points in C that are critical for the amoeba projection.

In general, the set cr(C) is a union of immersed circles in C. This classification problem admits a discriminant, a semi-algebraic hypersurface H in the space of curves C such that cr(C) is singular if and only if C is in H.

In this talk, we will review what is known about this classification problem away from the discriminant H and mention some open questions.

15:00-15:50 Annette Werner

Title: Tropical Geometry of the Hodge Bundle

Abstract: We study the image of the tropicalization map connecting the algebraic and the tropical Hodge bundle. For every pair consisting of a stable tropical curve G plus a divisor in the canonical linear system on G, we obtain a combinatorial condition to decide whether there is a smooth curve over a non-Archimedean field whose stable reduction has G as its dual graph together with an effective canonical divisor specializing to the given one. This is joint work with Martin Möller and Martin Ulirsch.

### Tuesday

9-9:50 Anders Björner

Title: Geometric lattices, discrete and continuous

10-10:50 Bernd Sturmfels

Title: The Geometry of Gaussoids

Abstract: Gaussoids offer a new link between combinatorics, statistics and algebraic geometry. Introduced by Lnenicka and Matus in 2007, their axioms describe conditional independence for Gaussian random variables. We explain this theory and how it relates to matroids. The role of the Grassmannian for matroids is now played by a projection of the Lagrangian Grassmannian. We discuss the classification and realizability of gaussoids, and we explore tropicalization, oriented gaussoids, and the analogue to positroids.

This is joint work with Tobias Boege, Alessio D'Ali and Thomas Kahle.

11-11:30 coffee

11:30-12:20 June Huh

Title: Kazhdan-Lusztig theory for matroids

Abstract: There is a remarkable parallel between the theory of Coxeter groups (think of the symmetric group Sn or the dihedral group Dn) and matroids (think of your favorite graph or point configuration). After giving an overview of the similarity, I will report on recent progress on the nonnegativity conjecture and the top-heavy conjecture for arbitrary matroids. Joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang.

12:30-14:00 lunch

14:00-14:20 Bo Lin

Title: The combinatorics of the tropical Hodge bundles

Abstract: The moduli space $M_g^{trop}$ of tropical curves of genus g is a generalized cone complex that parameterizes metric vertex-weighted graphs of genus g. For each such graph $\Gamma$, the associated canonical linear system $\vert K_\Gamma\vert$ has the structure of a polyhedral cell complex. We propose a tropical analogue of the Hodge bundle on $M_g^{trop}$ and study its combinatorial properties. Our construction is illustrated with explicit computations and examples.

This is a joint work with Martin Ulirsch.

14:30-14:50 Benjamin Schroeter

Title: The correlation constant of a field

Abstract: We discuss the basic question which appears in both the theory of graphs and finite geometries and ask how strongly independence of edges in a graph or vectors in a configuration is correlated. Here we follow the guideline of Huh and Wang and introduce as a measure a matroid and a field invariant; these are called the correlation constants of a matroid or field, respectively. It follows from one of their results that these constants are numbers in the interval from 0 to 2.We present an explicit matroid construction and show that the correlation constant of a field is at least $\frac{8}{7}$.

15:00-15:50 Kristin Shaw

Title: Bounding Betti numbers of patchworked real hypersurfaces by Hodge numbers

Abstract: The Smith-Thom inequality bounds the sum of the Betti numbers of a real algebraic variety by the sum of the Betti numbers of its complexification. In this talk I will explain our proof of a conjecture of Itenberg which, for a particular class of real algebraic projective hypersurfaces, bounds the individual Betti numbers in terms of the Hodge numbers of the complexification. The real hypersurfaces we consider arise from Viro’s patchworking construction, which is a powerful combinatorial method for constructing topological types of real algebraic varieties. To prove the bounds conjectured by Itenberg we develop a real analogue of tropical homology and use spectral sequences to compare it to the usual tropical homology of Itenberg, Katzarkov, Mikhalkin, Zharkov. Their homology theory gives the Hodge numbers of a complex projective variety from its tropicalization. Lurking in the spectral sequence of the proof are the keys to having combinatorial control of the topology of the real hypersurface produced from a patchwork.

This is joint work in progress with Arthur Renaudineau.

### Wednesday

9-9:50 Jonathan Wise

Title: Tropical data in algebraic moduli problems

Abstract: Logarithmic geometry serves as a bridge between tropical geometry and algebraic geometry. Using this connection, we can augment algebraic moduli problems by tropical data. I will describe some situations where this has been useful: a smooth, modular compactification of the space of genus 1 curves in projective space, and a modular resolution of the indeterminacy of the Abel-Jacobi map. These constructions are joint work with Steffen Marcus, Dhruv Ranganathan, and Keli Santos-Parker.

10-10:50 Kiran Kedlaya

Title: Tropical geometry of cluster varieties: a question from number theory

Abstract: We start with an elementary conjecture of David Robbins concerning the computation of determinants of p-adic matrices via the Dodgson-Jacobi method of condensation. This leads to a broader question, and some partial results, which can be formulated in terms of the tropical geometry of the varieties defined by cluster algebras. Joint work with Joe Buhler (CCR La Jolla).

11-11:30 coffee

11:30-12:20 Grigory Mikhalkin

Title: Examples of tropical-to-Lagrangian correspondence

Abstract: According to SYZ philosophy the same tropical object should admit two ways to be lifted classically: as a complex object, and as a (T-dual) symplectic object. While the tropical-to-complex correspondence is relatively well-studied, tropical-to-symplectic correspondence remains significantly less well-studied up to date.

In the talk we'll look at some first instances of this mirror correspondence between Lagrangian varieties and tropical cycles. As its application in the case of planar tropical curves we'll recover a theorem of Givental (from about 30 years ago) on Lagrangian embeddings of connected sums of Klein bottles to C^2.

For tropical curves in toric 3-folds the resulting Lagrangians turn out to be Waldhausen graph-manifolds. Rational curves of finite tropical multiplicity correspond to rational homology spheres among them. We show that the torsion in the first homology group of the Lagrangian 3-manifold is determined by the multiplicity of the tropical curve (in full compliance with Mirror Symmetry predictions).

12:30-14:00 lunch

14:00-14:20 Philipp Jell

Title: Smooth Mumford curves admit smooth tropicalizations

Abstract: Given a smooth algebraic curve X, it is natural to ask whether there exists an embedding of X into a toric variety such that the tropicalization of the image of X is a smooth tropical curve. More generally, does there exist a cofinal system of such embeddings?

I will report on work in progress which shows that the answer to both question is "Yes" for Mumford curves. Techniques involve extended skeleta in the sense of Baker, Payne and Rabinoff and a strengthening of their faithful tropicalization result.

14:30-14:50 Jens Forsgård

Title: Tropical Singularities

Abstract: Recently, the definition of the A-discriminant of Gelfand, Kapranov, and Zelevinsky was extended to real point configurations A. We briefly discuss the tropical A-discriminant in this setting, and singularities of tropical exponential sums. Our focus is on the relationship between the tropical discriminant and Esterov's iterated circuits.

15:00-15:50 Melody Chan

Title: Cohomology of M_g and the tropical moduli space of curves

Abstract: Joint with Søren Galatius and Sam Payne. We study the rational cohomology of M_g in degree 4g-6, showing that it grows exponentially in g. I will explain the relationship with the tropical moduli space of curves, and the combinatorial topology behind our result.

17:00 Reception at IML

### Thursday

9-9:50 Oleg Viro

Title: Non-tropical patchworking and three tropical kingdoms

10-10:50 Alex Fink

Title: Constructions of tropical ideals

Abstract: I will discuss some examples of tropical ideals, including a non-realisable tropical ideal structure for most hypersurfaces, and structural results on tropical ideals of zero-dimensional varieties which show that large classes of them are non-realisable and complicate the search for a finite way to present a tropical ideal.

This includes joint work with Jeff and Noah Giansiracusa and others.

11-11:30 coffee

11:30-12:20 Sam Payne

12:30-14:00 lunch

14:00-14:20 Kalina Mincheva

Title: The Picard group of a tropical toric scheme

Abstract: From any monoid scheme X (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) X_S by scalar extension to an idempotent semifield S. We prove that for a given irreducible monoid scheme X (satisfying some mild conditions) and an idempotent semifield S, the Picard group Pic(X) of X is stable under scalar extension to S (and to any field K). In other words, we show that the groups Pic(X) and Pic(X_S) (and Pic(X_K)) are isomorphic. In particular, if $X_\mathbb{C}$ is a toric variety, then Pic(X) is the same as the Picard group of the associated tropical scheme.

The Picard groups can be computed by considering the correct sheaf cohomology groups. We also construct the group CaCl(X_S) of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme X_S and prove that CaCl(X_S) is isomorphic to Pic(X_S).

14:30-15:20 Erwan Brugallé

Title: A generalization of Haas Theorem

Abstract: Viro's patchworking Theorem from the late 70's constitutes a powerful method to construct real algebraic hypersurfaces of toric varieties with a prescribed toplogy. In the case of plane curves, Haas classified in the 90's all combinatorial patchworkings that produce real algebraic curves with the maximal number of real components (aka $M$-curves). In this talk I will discuss an alternative interpretation of Haas Theorem in the light of the reformulation of combinatorial patchworking for plane curves in the tropical framework.

This is a joint work with Benoît Bertrand and Arthur Renaudineau.

15:30-15:50 Arthur Renaudineau

Title: Number of connected components of patchworked real plane curves

Abstract:I will describe the joint work with Kristin Shaw (see Kristin's abstract) in the particular case of plane curves. It turns out that in this case, the number of connected components of the real part is related to the dimension of the kernel of an explicit linear map. For example, using our techniques we provide another proof of Haas theorem describing maximal curves arising from patchworking.

### Friday

9-9:50 Diane Maclagan

Title: An update on tropical schemes

Abstract: A tropical scheme is described (locally) by a tropical ideal, which is an ideal in the semiring of tropical polynomials satisfying extra matroidal structure. I will review this definition, and give an update on recent progress in this area. Work of mine in this talk is joint with Felipe Rincon.

10-10:50 Ilia Itenberg

Title: Lines on quartic surfaces

Abstract: We study the possible values of the number of straight lines on a smooth surface of degree 4 in the 3-dimensional projective space. The question can be considered for various ground fields (or for the tropical semifield), the answer being depended on the choice made.

We show that the maximal number of real lines in a real non-singular spatial quartic surface is 56 (in the complex case, the maximal number is known to be 64). We also give a complete projective classification of non-singular complex quartics containing more than 52 lines: all such quartics are projectively rigid. Any value not exceeding 52 can appear as the number of lines of an appropriate complex quartic.

(Joint work with A. Degtyarev and A. S. Sertöz.)

11-11:30 coffee

11:30-12:20 Renzo Cavalieri

Title: On the generating function for Mumford's kappa classes

Abstract: Kappa classes were introduced by Mumford, as a tool to explore the intersection theory of the moduli space of curves. Iterated use of the projection formula shows there is a close connection between the intersection theory of kappa classes on the moduli space of unpointed curves, and the intersection theory of psi classes on all moduli spaces. In terms of generating functions, we show that the potential for kappa classes is related to the Gromov-Witten potential of a point via a change of variables essentially given by complete symmetric polynomials. Surprisingly, the starting point of this story is a combinatorial formula that relates intersections of kappa classes and psi classes via a graph theoretic algorithm (the relevant graphs being dual graphs to stable curves).This is joint work (in progress) with Vance Blankers.

### Program Committee:

Sandra Di Rocco - *KTH Royal Institute of Technology*

Jan Draisma - *University of Bern*

Anders Jensen - *Aarhus University*

Hannah Markwig - *Universität Tübingen*

Benjamin Nill - *Otto von Guericke Universität Magdeburg*