Related papers: $PSL_2$ tropicalization and lines on surfaces
In this thesis we study toric degenerations of projective varieties. We compare different constructions to understand how and why they are related as s first step towards developing a global framework. In focus are toric degenerations…
Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model $\mathscr X$ of $X$ plus a suitable horizontal…
In this paper, we perform a detailed investigation on the various geometrical properties of trapped surfaces and the boundaries of trapped region in general relativity. This treatment extends earlier work on LRS II spacetimes to a general 4…
In this paper we consider the classification of minimal cellular structures of spaces of topological complexity two under some hypotheses on there graded cohomological algebra. This continues the method used by M.Grant et al. in [1].
We consider the stable ruled surface $S_1$ over an elliptic curve. There is a unique foliation on $S_1$ transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the…
We recast elliptic surfaces over the projective line in terms of the non-commutative tori and one-parameter families of the periodic continued fractions. The correspondence is used to study the Picard numbers, the ranks and the minimal…
In this paper we present an algorithm for computing a matrix representation for a surface in P^3 parametrized over a 2-dimensional toric variety T. This algorithm follows the ideas of [Botbol-Dickenstein-Dohm-09] and it was implemented in…
Let T be a split torus over local or global function field. The theory of Brylinski-Deligne gives rise to the metaplectic central extensions of T by a finite cyclic group. The representation theory of these metaplectic tori has been…
In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the…
In this article we study the tropicalization of the Hilbert scheme and its suitability as a parameter space for tropical varieties. We prove that the points of the tropicalization of the Hilbert scheme have a tropical variety naturally…
These notes were intended as support material for a minicourse on Anosov flows in the conference "Symplectic geometry and Anosov flows'' which took place in Heidelberg in July 2024 organized by Peter Albers, Jonathan Bowden and Agust\'in…
The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is…
We present a new normal form for cubic surfaces that is well suited for p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27 trees in the tropicalization. The new normal form is a polynomial with eight terms,…
I discuss the general formalism of two-dimensional topological field theories defined on open-closed oriented Riemann surfaces, starting from an extension of Segal's geometric axioms. Exploiting the topological sewing constraints allows for…
Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical…
For \ell \neq p odd primes, we examine PSL_2(\ell)-covers of the projective line branched at one point over an algebraically closed field of characteristic p, where PSL_2(\ell) has order divisible by p. We show that such covers can be…
Recently, a theory for the tropicalization of a spherical homogeneous space $G/H$ was developed by Tassos Vogiannou. We extend his ideas to define the tropicalization of a spherical $G/H$-embedding. This generalizes the construction of…
Tropical geometry and the theory of Newton-Okounkov bodies are two methods which produce toric degenerations of an irreducible complex projective variety. Kaveh-Manon showed that the two are related. We give geometric maps between the…
Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties,…
We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. We show that this property depends…