Related papers: Smooth tropical surfaces with infinitely many trop…
We present applications of tropical geometry to some integrable piecewise-linear maps, based on the lecture given by one of the authors (R. I.) at the workshop "Tropical Geometry and Integrable Systems" (University of Glasgow, July 2011),…
We show that the pure mapping class group is uniformly perfect for a certain class of infinite type surfaces with noncompact boundary components. We then combine this result with recent work in the remaining cases to give a complete…
We investigate three-dimensional surfaces where the normal vector forms a constant angle with the radius vector. These surfaces naturally extend equiangular (logarithmic) spirals in the plane.
We study rationality properties of smooth complete intersections of three quadrics in $\mathbb{P}^7$. We exhibit a smooth family of such intersections with both rational and non-rational fibers.
We solve the following geometric problem, which arises in several three-dimensional applications in computational geometry: For which arrangements of two lines and two spheres in R^3 are there infinitely many lines simultaneously…
Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In…
A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal…
Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive class [f_p] in E(1)_p when p>1. As a…
We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.
In this paper we define $q$-spherical surfaces as the surfaces that contain the absolute conic of the Euclidean space as a $q-$fold curve. Particular attention is paid to the surfaces with singular points of the highest order. Two classes…
This thesis delves into the geometry of abstract tropical curves, exploring their complete linear system and associated tropical submodules. We establish a lower bound on the dimension of tropical submodules in terms of the Baker-Norine…
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…
We first review some topics in the classical computational geometry of lines, in particular the O(n^{3+\epsilon}) bounds for the combinatorial complexity of the set of lines in R^3 interacting with $n$ objects of fixed description…
We enumerate the number of surfaces of degree $d$ in $P^3$ having a singular line of order $k$, passing through $\delta$ generic points (where $\delta$ is the dimension of moduli space of such surfaces).
Over a field k of characteristic 3, we prove that there are no smooth quartic surfaces S in IP^3 with more than 112 lines. Moreover, the surface with 112 lines is projectively equivalent over k-bar to the Fermat quartic. As a key…
We prove that a K3 quartic surface defined over a field of characteristic 2 can contain at most 68 lines. If it contains 68 lines, then it is projectively equivalent to a member of a 1-dimensional family found by Rams and Sch\"utt.
A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain…
We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental…
In the present paper we study two-dimensional maximal surfaces with harmonic level-sets. As a corollary we obtain a new class of one-periodic maximal surfaces.
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated…