Related papers: Classifying toric surface codes of dimension $7$
In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we define two convex-geometric notions: the Q-codegree and the nef value of a…
Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms…
From a finite set in a lattice, we can define a toric variety embedded in a projective space. In this paper, we give a combinatorial description of the dual defect of the toric variety using the structure of the finite set as a Cayley sum…
Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…
This paper focuses on the classification of all toric log Del Pezzo surfaces with exactly one singularity up to isomorphism, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective…
The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in…
We construct algebraic geometric codes from del Pezzo surfaces and focus on the ones having Picard rank one and the codes associated to the anticanonical class. We give explicit constructions of del Pezzo surfaces of degree 4, 5 and 6,…
In this expository note, we review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for K3 surfaces which relies on a sublattice…
In this paper we deduce the Lebesgue and the Knaster--Kuratowski--Mazurkiewicz theorems on the covering dimension, as well as their certain generalizations, from some simple facts of toric geometry. This provides a new point of view on this…
Subspace codes have received an increasing interest recently due to their application in error-correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and…
The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical…
We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.
Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms. They have been studied for decades and a lot of progress has been made.…
Surface and color codes are two forms of topological quantum error correction in two spatial dimensions with complementary properties. Surface codes have lower-depth error detection circuits and well-developed decoders to interpret and…
For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous…
In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors of different order that encode information of the path in a compact form. When the path varies, such signatures…
The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^*$-polynomials of lattice polytropes, which…
In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus…
A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the…
We give a bound of $k$ for a very ample lattice polytope to be $k$-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties.