Related papers: Differential Goppa Codes

The paper introduces a new differential-geometric system which originates from the theory of $m$-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise…

We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we…

We introduce the first geometric construction of codes in the sum-rank metric, which we called linearized Algebraic Geometry codes, using quotients of the ring of Ore polynomials with coefficients in the function field of an algebraic…

In this paper we construct a geometric model for the triangulated category generated by the simple modules of any graded gentle algebra. This leads to a geometric model of their perfect derived categories and by a recent paper of Booth,…

Let $p$ be a prime number such that the modular curve $X_0(p)$ has genus at least two. We show that the only points of the reduction mod $p$ of $X_0(p)$ with image in the reduction mod $p$ of $J_0(p)$ in the cuspidal group are the two…

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local…

In this work, we study the classification of differential graded algebras over $\mathbb{Z}$ (DGAs) whose homology is $\mathbb{F}_p[x]$, i.e. the polynomial algebra over $\mathbb{F}_p$ on a single generator. This classification problem was…

We define a differential graded algebra associated to Legendrian knots in thickened convex surfaces $\Sigma\times \mathbb{R}$. The algebra is defined in the same spirit as the Chekanov-Eliashberg DGA for Legendrians in $\mathbb{R}^3$, but…

This paper presents explicit constructions of bases for Riemann-Roch spaces associated with arbitrary divisors on elliptic curves. In the context of algebraic geometry codes, the knowledge of an explicit basis for arbitrary divisors is…

Spherical codes, with a rich history spanning nearly five centuries, remain an area of active mathematical exploration and are far from being fully understood. These codes, which arise naturally in problems of geometry, combinatorics, and…

Let $\gamma_1,\gamma_2$ be a pair of constant-degree irreducible algebraic curves in $\mathbb{R}^d$. Assume that $\gamma_i$ is neither contained in a hyperplane nor in a quadric surface in $\mathbb{R}^d$, for each $i=1,2$. We show that for…

The Quot scheme of points $\mathrm{Quot}_{d,n}(X)$ on a variety $X$ over a field $k$ parametrizes quotient sheaves of $\mathcal{O}_X^{\oplus d}$ of zero-dimensional support and length $n$. It is a rank-$d$ generalization of the Hilbert…

We study generalised differential structures $\Omega^1,d$ on an algebra $A$, where $A\tens A\to \Omega^1$ given by $a\tens b\to a d b$ need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf…

We study geometric variations of the discriminating code problem. In the \emph{discrete version} of the problem, a finite set of points $P$ and a finite set of objects $S$ are given in $\mathbb{R}^d$. The objective is to choose a subset…

Let $n_0, n_1, \ldots, n_p$ be a sequence of positive integers such that $n_0 < n_1 < \cdots < n_p$ and $\mathrm{gcd}(n_0,n_1, \ldots,n_p) = 1$. Let $S = \langle (0,n_p), (n_0,n_p-n_0),\ldots,(n_{p-1},n_p-n_{p-1}), (n_p,0) \rangle$ be an…

We consider linear error correcting codes associated to higher dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult…

We produce a flexible tool for contracting subcurves of logarithmic hyperelliptic curves, which is local around the subcurve and commutes with arbitrary base-change. As an application, we prove that hyperelliptic multiscale differentials…

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary…

In this paper we treat several topics regarding numerical Weierstrass semigroups and the theory of Algebraic Geometric Codes associated to a pair $(X, P)$, where $X$ is a projective curve defined over the algebraic closure of the finite…

The geometric models for the module category and derived category of any gentle algebra were introduced to realize the objects in module category and derived category by permissible curves and admissible curves respectively. The present…