Related papers: Tropical elliptic curves in 3-space
The paper is devoted to finding the colorings of the edges of the 1-skeleton of triangulations of the 2-sphere in three colors so that for each face all three of its sides have different colors. First, by the method of adding one vertex…
Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in 3-space. These are balanced rational…
Let $X$ be an algebraic variety and let $S$ be a tropical variety associated to $X$. We study the tropicalization map from the moduli space of stable maps into $X$ to the moduli space of tropical curves in $S$. We prove that it is a…
Given a principally polarized abelian variety $(A,\Theta)$, we give a characterization of all elliptic curves that lie on $A$ in terms of intersection numbers of divisor classes in its N\'eron-Severi group.
There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already…
We study the number of elliptic curves, up to isomorphism, over a fixed quartic field $K$ having a prescribed torsion group $T$ as a subgroup. Let $T=\Z/m\Z \oplus \Z/n\Z$, where $m|n$, be a torsion group such that the modular curve…
We provide a complete classification of possible configurations of mutually pairwise touching infinite cylinders in Euclidian 3D space. It turns out that there is a maximum number of such cylinders possible in 3D independently on the shape…
Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler class and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This…
We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…
We study Galois embedding problems arising from the 3-torsion of elliptic curves defined over $\mathbb{Q}$, extending the correspondence to all possible images of mod 3 Galois representations; namely,…
A degeneration of a singular curve on a toric surface, called a tropicalization, was constructed by E. Shustin. He classified the degeneration of 1-cuspidal curves using polyhedral complexes called tropical curves. In this paper, we define…
We develop a technique to study curves in a variety which has a degeneration into some union of varieties. The class of such varieties is very broad, but the theory becomes particularly useful when the variety has a degeneration into a…
The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$…
We carry out a complete analysis of the toric elliptic and genus-one fibrations of all 474 million reflexive polytopes in the Kreuzer-Skarke database. Earlier work with Huang showed that all but 29,223 of these polytopes have such a…
We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.
We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant…
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in the $\mathbb{Q}$-isogeny class…
We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work…
We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the only 3-connected,…