Related papers: On algebraic space filling curves

We investigate minimal degree smooth algebraic space filling curves on the product of projective lines. We prove that there are plenty of examples in an explicit sense, extending the existence result of Homma and Kim.

A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth…

The group PGL(3) of linear transformations of the projective plane acts naturally on the projective space parametrizing curves of a given degree. In this note we begin the study of the orbits of smooth curves under this action: we construct…

A theorem of Wiegerinck asserts that the Bergman space of an open subset of the complex numbers is either infinite-dimensional or trivial. Recently, this has been generalized to holomorphic vector bundles over the projective line by the…

We study projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth…

Let $X$ be a smooth algebraic curve. Suppose that there exists a triple covering $f : X \to Y$ where $Y$ is a smooth algebraic curve. In this paper, we investigate the existence of morphisms from $X$ to the projective line $\mathbf{P}^1$…

We show that for any simple closed curve in the sphere at infinity of a Gromov hyperbolic 3-space with cocompact metric, there exist a properly embedded least area plane in the space spanning the given curve. This gives a positive answer to…

In this paper we prove that, for any $n\ge 3$, there exist infinitely many $r\in \N$ and for each of them a smooth, connected curve $C_r$ in $\P^r$ such that $C_r$ lies on exactly $n$ irreducible components of the Hilbert scheme…

Given a smooth complex variety $X$, an algebraically skew embedding of $X$ is an embedding of $X$ into a complex projective space $\mathbb{P}^N$ such that for any two points $x,y\in X$, their embedded tangent spaces in $\mathbb{P}^N$ do not…

We study plane curves over finite fields whose tangent lines at smooth $\mathbb{F}_q$-points together cover all the points of $\mathbb{P}^2(\mathbb{F}_q)$.

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by R\'edei-type polynomials, are powerful in studying blocking sets. In this paper, we…

We give an algebraic construction of the moduli space of irregular singular connections of generic ramified type on a smooth projective curve. We prove that the moduli space is smooth and give its dimension. Under the assumption that the…

We use Poonen's closed point sieve to prove two independent results. First, we show that the obvious obstruction to embedding a curve in a smooth surface is the only obstruction over a perfect field, by proving the finite field analogue of…

We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere,…

We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface, a projective homogeneous…

We show the existence of a hypersurface that contains a given closed subscheme of a projective space over a finite field and intersects a smooth quasi-projective scheme smoothly, under some condition on the dimension. This generalizes a…

Hilbert's two-dimensional space-filling curve is appreciated for its good locality properties for many applications. However, it is not clear what is the best way to generalize this curve to filling higher-dimensional spaces. We argue that…

We present a framework for constructing examples of smooth projective curves over number fields with explicitly given elements in their second K-group using elementary algebraic geometry. This leads to new examples for hyperelliptic curves…

We describe a search for plane-filling curves traversing all edges of a grid once. The curves are given by Lindenmayer systems with only one non-constant letter. All such curves for small orders on three grids have been found. For all…