Workshop NOTrop - Titles and abstracts

Lara Bossinger

Title: Towards a fundamental theorem for totally positive tropical geometry

Abstract: The fundamental Theorem of (algebraic) tropical geometry states that three different approaches to Tropicalization yield the same tropical variety. However, the different notions of tropicalzation treated in the theorem are not exhaustive: for schemes with a positive atlas Fock and Goncharov introduced another tropicalzation as functor of points over a semi field. In this talk I will explain Fock—Goncharov’s tropicalization and show that it fits into a positive analogue of the fundamental theorem extending results of Speyer and Williams. The result holds for cluster varieties of finite type that arise naturally in this context. The talk is based on results in arxiv:2208.01723.

Ian Cavey

Title: Hilbert schemes and Newton-Okounkov bodies

Abstract: The Hilbert scheme of n points on a smooth (projective) surface S is a smooth (projective) variety that parametrizes finite, closed subschemes of S with length n. The Picard groups of these Hilbert schemes are known, but in most cases the global sections of the line bundles are poorly understood. In this talk, I will show how these spaces of global sections can be studied via Newton-Okounkov bodies in the case that S is a toric surface. I will give an exact description of the Newton-Okounkov bodies of the Hilbert schemes of points in the affine plane. I will also give a conjecturally sharp upper bound for the Newton-Okounkov bodies of Hilbert schemes of points in some projective toric surfaces, which are built from the Newton polygons of line bundles on the underlying surface.

Alheydis Geiger

Title: Self-dual matroids from canonical curves

Abstract:

A hyperplane section of a canonically embedded, non-hypereliptic smooth curve of genus g consists of 2g-2 points in g-2 dimensional projective space. From work by Dolgachev and Ortland it is known that point configurations obtained in such a way are self-associated. We interpret this notion in terms of matroids: A generic hyperplane section of a canonical curve gives rise to an identically self-dual matroid. This can also be seen as a combinatorial shadow of the Riemann-Roch theorem. This procedure also works for graph curves, which are stable canonical curves consisting of 2g-2. Self-dual point configurations are parametrized by a subvariety of the Grassmannian Gr(n,2n) and its tropicalization. We provide an extensive analysis of the tropical self-dual Grassmannian trop(SGr(3,6)) and pave the way for a thorough investigation of tropical Cayley Octads. Further, we classify identically self-dual matroids of rank up to five and determine the dimension of their (self-dual) realization spaces. Building on work by Petrakiev, we investigate the question, which self-dual point configurations can be obtained by a hyperplane section with a canonical curve. This project is work in progress with Sachi Hashimoto, Bernd Sturmfels and Raluca Vlad.

Milena Hering

Title: Equations of toric vector bundle

Abstract: The projectivisation of a very ample toric vector bundle admits a natural embedding into projective space. We describe defining equations for this embedding in a larger projective space via a more natural embedding of the vector bundle in the Cox ring of a toric variety. This is joint work with Diane Maclagan and Greg Smith.

Nathan Ilten

Title: Khovanskii-finiteness for rational curves of genus two and low degree hypersurfaces

Abstract: Given a projective variety $X$, is there a homogeneous full-rank valuation on its homogeneous coordinate ring such that the value semigroup is finitely generated? This is a subtle question with connections to tropical geometry, Newton-Okounkov theory, and toric degenerations. In this talk, I will discuss two situations in which we can give at least partial answers. Firstly, for (singular) rational curves of arithmetic genus two, we give an effective method for answering this question. Secondly, we show that for $X$ a general hypersurface in $\mathbb{P}^n$ of degree at most $2n-1$, there is a homogeneous full-rank valuation whose value semigroup is (finitely) generated by elements of degree one. This is joint work with Ahmad Mokhtar and Oscar Lautsch.

Hernan Iriarte

Title: Geometry with higher rank valuations

Abstract: Motivated by the study of multi-stage degenerations and the theory of Okounkov bodies, we introduce new tools to work with objects from algebraic geometry using higher rank valuations. From one part, we obtain higher rank analogs of the notion of skeleta on Thuiller's analytification of algebraic varieties. Their points correspond to tangent directions on the dual cone complex and they act as partial derivative operators on rational functions. The new structures involved can be regarded as "polyhedra over the ordered ring R[x]/(x^k)". This motivates us to study the polyhedral geometry over this ring, which shares many properties with the usual polyhedral geometry over R but has a different combinatorics. This new polyhedral geometry allows us to study more general instances of tropical and analytic geometry of higher rank which we expect to play an important role in the study of multiparameter degenerations.

Kiumars Kaveh

Title: Degenerations of polarized projective varieties to complexity-one varieties

Abstract: It is known (by a result of Dave Anderson) that if the value semigroup of a (full rank) valuation on the homogeneous coordinate ring (of a projective variety) is finitely generated then the projective variety has a flat degeneration to a (possibly non-normal) toric variety. It is an important problem to determine if a given homogeneous coordinate ring admits a (full rank) valuation with finitely generated semigroup (and hence a toric degeneration). More recently, Takuya Murata, Chris Manon and the speaker show that ANY embedded projective variety has an "almost" toric degeneration in the following sense: ANY homogeneous coordinate ring (of a projective variety X) admits a valuation of rank dim X - 1 such that the corresponding associated graded is finitely generated and hence X can be flatly degenerated to the corresponding complexity-one T-variety.

We discuss this results in the talk as well as some results of Ilten-Wrobel in this regard. We also mention examples of embedded projective curves that do not admit any toric degenerations.

Peter Littelmann

Title: On Seshadri stratifications, its Newton–Okounkov complexes and semi-toric degenerations.

Abstract: In the talk I will introduce the notion of a Seshadri stratification on an embedded projective variety.

Such a structure allows us to associate to the variety a Newton-Okounkov simplicial complex with

an extra integral structure, and a flat degeneration of the variety into a reduced union of toric varieties.

For Schubert varieties for example, combinatorial objects like the Lakshmibai-Seshadri paths get

an interpretation as successive vanishing orders of regular functions.

This is on joint work with Xin Fang (Cologne) and Rocco Chirivì (Lecce).

Hannah Markwig

Title: A new perspective on tropical covers

Abstract: Hurwitz numbers count covers of Riemann surfaces with prescribed ramification profiles. For covers of the Riemann sphere with two special ramification profiles and only simple branch points else, these numbers can be determined completely combinatorially in terms of counting tropical covers. We give a new interpretation of this count as an intersection product with the tropical double ramification cycle. This perspective can be generalized to the pluricanonical double ramification cycle, yielding a count of tropical leaky covers which miss the balancing condition.

Joint work with Renzo Cavalieri and Dhruv Ranganathan

Mirko Mauri

Title: Index of log Calabi-Yau pairs with maximal intersections

Abstract: A Calabi-Yau variety is a variety whose canonical bundle has degree zero along all its curves. It is known that in this case a multiple of the canonical bundle is a trivial line bundle and the index of the variety is the smallest such multiple. Esser, Totaro and Wang have recently constructed examples of Calabi-Yau varieties whose index grows doubly exponentially with the dimension. We study the same problem for log Calabi-Yau pairs $(X,D)$, i.e. a multiple of $K_X+D$ is trivial. Our result however goes in the opposite direction: the index of the pair $(X,D)$ is at most 2 independently of the dimension $d$, provided that $D$ is reduced and has maximal intersection, i.e. $d$ components of $D$ intersect at one point. We explain it in terms of the orientability of dual complexes and/or the residues of 1-forms on $\mathbb{P}^1$. I will discuss also some applications of these results in mirror symmetry.

Johannes Nicaise

Title: Tropical degenerations and irrationality of hypersurfaces in products of projective spaces

Abstract: The specialization map for stable birational types associates with every strictly toroidal one-parameter degeneration an obstruction to the stable rationality of a very general fiber. In many applications, suitable degenerations can be constructed by hand, but there are also cases where the complexity gets too high to write down explicit equations, and one needs to rely on a combinatorial description in terms of regular subdivisions of Newton polytopes. I will illustrate this technique for hypersurfaces in products of projective spaces. This is based on joint work with John Christian Ottem.

Joaquim Roé

Title: Irrational nef rays at the boundary of the Mori cone for very general blowups of the plane.

Abstract: We will discuss a technique for discovering irrational rays at the boundary of the Mori cone for linear systems on a general blowup of the plane, and give examples of such irrational rays. This is joint work with Ciro Ciliberto and Rick Miranda

Raman Sanyal

Title: Flag polymatroids

Abstract: Abstract:

The well-known greedy algorithm is a mean for optimizing over matroids. Geometrically, the greedy algorithm traces a monotone path in the graph of the associated independent set polytope. It turns out that all monotone paths are greedy paths and all are coherent in the sense of Billera-Sturmfels. Thus, monotone paths are parametrized by the associated monotone path polytope. We show that this monotone path polytope is essentially a flag matroid of Gelfand et al. and the construction hints at a curious connection between torus-orbit closures in Grassmannians and complete flag varieties. In this talk I will explain the polyhedral geometry behind all this from the more general perspective of polymatoids. This is joint work with Alex Black.

Rob Silversmith

Title: Cross-ratios and perfect matchings

Abstract: Given a bipartite graph G (subject to a 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. I’ll also outline the tropical side of the story.

Stefano Urbinati

Title: Mori Dream Pairs and C^*-actions

Abstract: In this talk I am presenting a joint work with Lorenzo Barban, Eleonora A. Romano and Luis E. Solá Conde.

The idea of the talk is that of giving a connection between 'local' Mori theory and C^*actions. In particular, we construct and characterize a correspondence between Mori dream regions arising from small modifications of normal projective varieties and C^*actions on polarized pairs which are bordisms.

Claudia Yun

Title: Discrete Morse Theory for Symmetric Delta-complexes

Abstract: In this talk, we generalize discrete Morse theory to the context of symmetric Delta-complexes. First introduced by Forman, discrete Morse theory is an adaptation of Morse theory for regular CW-complexes. It gives a schematic for collapsing cells in a CW-complex without changing its homotopy type. On the other hand, symmetric Delta-complexes are a generalization of cell-complexes. They are topological spaces built from quotients of standard simplices, and they have played important roles in recent developments of tropical geometry. We generalize various concepts from discrete Morse theory, including discrete Morse functions and acyclic matchings on face posets, and prove parallel theorems for symmetric Delta-complexes. We also apply this new theory to the moduli space of tropical abelian varieties.

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