Related papers: Tropical degenerations and stable rationality
In this article it is proved that the dynamical properties of a broad class of semilinear parabolic problems are sensitive to arbitrarily small but smooth perturbations of the nonlinear term, when the spatial dimension is either equal to…
We study semi-stable degenerations of toric varieties determined by certain partitions of their moment polytopes. Analyzing their defining equations we prove a property of uniqueness.
The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials universally supported on degeneracy loci;…
We consider a toric degeneration of Calabi--Yau complete intersections of Batyrev--Borisov in the Gross--Siebert program. One can associate two types of tropical spaces with it. One is a tropical variety obtained by tropicalization. The…
We define an intersection product of tropical cycles on matroid varieties (via cutting out the diagonal) and show that it is well-behaved. In particular, this enables us to intersect cycles on moduli spaces of tropical rational marked…
Tropical geometry is a degeneration of classical geometry which loose the property of unique factorization for polynomials. In this paper we explore a structure that is known to be a semi-degeneration between the classical algebra and the…
We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.
We prove the existence of a family $\mathcal{X}\rightarrow B$ of smooth projective fourfolds, such that the very general fiber $\mathcal{X}_t$ is not stably rational (a fortiori not rational), but some special fibers $\mathcal{X}_t$ are…
We explore extensions of tropical methods to arithmetic enumerative problems such as $\mathbb{A}^1$-enumeration with values in the Grothendieck-Witt ring, and rationality over Henselian valued fields, using bitangents to plane quartics as a…
We prove the birational superrigidity and nonrationality of a hypersurface in $\mathbb{P}^{6}$ of degree 6 having at most isolated ordinary double points.
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure…
We investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to…
We construct closed immersions from initial degenerations of the spinor variety $\mathbb{S}_n$ to inverse limits of strata associated to even $\Delta$-matroids. As an application, we prove that these initial degenerations are smooth and…
Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for cubic graphs. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless cubic…
Two cubic equations and three auxiliary equations for edges and face diagonals of a rational perfect cuboid have been recently derived. They constitute a background for two inverse problems. The coefficients of cubic equations and the right…
We study the classical result by Bruijn and Erd\H os regarding the bound on the number of lines determined by a $n$-point configuration in the plane, and in the light of the recently proven Tropical Sylvester-Gallai theorem, come up with a…
We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this…
Richmond and Richmond (Amer. Math. Monthly 104 (1997), 713--719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove…
Some classes of cubic fourfolds are birational to fibrations over $P^2$, where the fibers are rational surfaces. This is the case for cubics containing a plane (resp. an elliptic ruled surface), where the fibers are quadric surfaces (resp.…
We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations…