Schedule for: 19w5100 - Tropical Methods in Real Algebraic Geometry

Real tropical varieties are polyhedral objects which are in some cases isotopic to real algebraic varieties. We will introduce those objects and show an upper bound on their Betti numbers. These bounds are given in terms of the dimension of tropical homology groups modulo 2 of the underlying tropical variety, and are derived from a certain spectral sequence. In the case of hypersurfaces, we prove that tropical homology groups are torsion free, implying a bound conjectured by Itenberg on the Betti numbers of the real part of a real hypersurface near the tropical limit in terms of Hodge numbers of the complexification. This is a joint work with Kristin Shaw, and with Charles Arnal and Kristin Shaw.

Very recently, Oleg Viro has introduced "round-dances of points" on real curves. Earlier, Mario Kummer and Kristin Shaw have already shown that all type I curves admit non-trivial dances. We'll explore implications of such dances to geometry of real canonical divisors.

I will define and discuss the tropicalization and analytification of semialgebraic sets. We show that the real analytification is homeomorphic to the inverse limit of real tropicalizations, analogously to a result of Payne. We also show a real analogue of the fundamental theorem of tropical geometry. This is based on joint work with Philipp Jell and Claus Scheiderer.

The first example of a nonnegative polynomial which is not a sum of squares was Motzkin's polynomial $g(x,y) = x^4 y^2 + x^2 y^2 - 3 x^2 y^2 + 1$. This is an example of an agiform, that is, a polynomial which can be obtained from the inequality of arithmetic and geometric means by a monomial substitution. Given a support set A, each simplicial circuit contained in A gives rise to a family of agiforms. In the fewnomial approach to nonnegativity, we study the cone generated by all agiforms, called the sonc-cone. In this talk, we will describe the boundary of the sonc-cone as a stratified space. As it turns out, each strata is parametrized by a family of tropical hypersurfaces.

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.

Using tools from Tropical and Non-Archimedean Geometry, we show that there is a tight relationship between the following two concepts of real inflection of real linear series defined on real algebraic curves: 1. that of complete series on hyper-elliptic curves, and 2. that of incomplete series on elliptic curves. Concretely, the case (1) can be degenerated to the case (2), and the case (2) can be regenerated to the case (1). This interplay gives us two products: 1. A limit linear series on a (marked) metrized complex of (real) algebraic curves. By this we mean a marked tropical curve with real models. 2. A 2-dimensional family of polynomials generalizing the division polynomials (which are used to compute the torsion points of elliptic curves) This is a joint work with I. Biswas (TATA, India) and E. Cotterill (UFF, Brazil).

I will discuss an ongoing project with John Christian Ottem to find new examples of stably irrational hypersurfaces by combining the specialization results obtained with Evgeny Shinder with tropical compactification techniques. The guiding example will be the quartic fivefold.

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves with a fixed degree in the real projective plane is a classical subject that has undergone considerable evolution. On the other hand, apart from studies concerning Hirzebruch surfaces and at most degree 3 surfaces in the real projective space, not much is known for more general ambient surfaces. In particular, this is because varieties constructed using the patchworking method are hypersurfaces of toric varieties. However, there are many other real algebraic surfaces. Among these are the real rational surfaces, and more particularly the real minimal rational surfaces. In this talk, we present some results about the classification of topological types realized by real algebraic curves of "small class" in real minimal del Pezzo surfaces which are real non-toric surfaces with non-connected real parts. We will explain how combine variations of classical methods with degeneration methods, that have found recent applications in real enumerative geometry, and the exploitation of Welschinger invariants to get through such classifications.

In this talk, I will introduce the notion of preorder on a group and give some examples. Then I will talk about some classical topologies on the set $ZR(G)$ of preorders on a group $G$ and I will finish giving a compactness result of $ZR(G)$ for all the classical topologies. This is a joint work with Guillaume Rond.

In this talk we study a tropical enumerative problem which is a generalization of the the planar problem dealing with curves meeting infinity in fixed points and studied by Mikhalkin. We show that it admits a quantization.

A versal deformation of a real plane curve singularity contains the so-called Severi loci parameterizing deformations with a given total delta-invariant (and among them the equigeneric stratum), and the cuspidal equigeneric loci (among them the equiclassical stratum). The multiplicities of these analytic spaces germs can be regarded as local analogues of Gromov-Witten invariants. We show that some of them possess real multiplicities, which can be viewed as local analogues of Welschinger invariants. We also show that some of these real invariants can be converted to global invariants counting real rational curves having prescribed number of cusps and belonging to a suitable linear system on a real toric surface. We also discuss ways to compute the considered invariants via tropical geometry.

In this talk I will introduce the classical and tropical invariants corresponding to the enumeration of fixed genus (or cogenus) curves on a Surface, the floor diagrams, and their multiplicity in order to explicit some combinatorial aspects that allow us to establish the polynomiality property on the degree. I will explain how these polynomials can be seen as polynomials in two variables, the degree and the number of complex conjugated points of the configuration of points; and I will show some properties of the polynomials we obtain and some relations among them coming from known properties of the classical invariants. This is a joint work with E. Brugallé.

Let $U$ be the space of irreducible nodal rational curves of a given degree in a given toric surface. We investigate how we can permute the nodes of such a curve by traveling along loops in the space $U$, that is we want to determine the image of a certain monodromy map. Using elementary considerations on amoebas of Harnack curves, we give a description of the image of the latter monodromy in a fairly general situation.

We examine some phenomena of total reality in real algebraic geometry for topological and geometric aspects. This will also lead to some questions that might be approached with tropical methods.

J. Solomon and S. Tukachinsky constructed open Gromov-Witten invariants in their 2016 papers from an algebraic perspective of $A_{\infty}$-algebras of differential forms. We present a geometric translation of their construction. In this geometric perspective, the resulting invariants readily reduce to Welschinger's open invariants of symplectic sixfolds, which count multi-disks weighted by the linking numbers between their boundaries. Solomon-Tukachinsky's open WDVV-relations and their proof can be translated similarly.

The homogeneous coordinate ring of the Grassmannian (with respect to its Pluecker embedding) has the structure of a cluster algebras (due to Scott). In this talk I will explain for $Gr(2,n)$ equipping this cluster algebra with Fomin-Zeleviskys universal coefficients gives rise to a (closed) maximal cone in the Groebner fan of the Pluecker ideal. Intersecting the tropical Grassmannian with this maximal Groebner cone yields the totally positive part of the tropicalization. (Joint work with Fatemeh Mohammadi and Alfredo Nájera Chávez)

There are $280$ binodal cubic surfaces passing through $17$ general points. We count that for the typically used tropical point conditions $214$ of these give tropicalizations such that the nodes are separated on the tropical cubic surface. This is joint work with Alheydis Geiger.

We show that the cycle class map from the Chow ring of a matroid to the tropical cohomology groups of the wonderful compactification of the Bergman fan induces an isomorphism of rings. We then discuss connection to the work of Adiprasito, Huh and Katz on log-concavitiy of the coefficients of the characteristic polynomials. Joint work with Matthieu Piquerez.

We discuss possible numbers of 2-planes in a smooth cubic hypersurface in the 5-dimensional projective space. We show that, in the complex case, the maximal number of planes is 405, the maximum being realized by the Fermat cubic. In the real case, the maximal number of planes is 357. The proofs deal with the period spaces of cubic hypersurfaces in the 5-dimensional complex projective space and are based on the global Torelli theorem and the surjectivity of the period map for these hypersurfaces, as well as on Nikulin's theory of discriminant forms. Joint work with Alex Degtyarev and John Christian Ottem.

Given the amoeba X of an algebraic variety, we are interested in finding a tropical variety of the same dimension, degree, etc. which approximates X in a certain sense, called the/a spine of X. The name originates in the work of Passare-Rullgard who constructed and studied spines in the case of hypersurfaces using the so-called Ronkin function. In my talk, I will focus on curves, instead. In this case, for some applications it is necessary for the curve and its spine to have the same genus. We show that any for any degree D there is a bound B(D) such that for any amoeba X of a rational curve of that degree there exists a rational tropical curve of the same degree and within Hausdorff-distance B(D) to X. It is interesting to note that even in the case of plane curves the Passare-Rullgard spine does not have the correct genus. In the remaining time, I will present some further questions and possible approaches in this direction. (joint with Grigory Mikhalkin)