Related papers: On free curves and related open problems
Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries…
A new concept called multilevel contours is introduced through this article by the author. Theorems on contours constructed on a bundle of complex planes are stated and proved. Multilevel contours can transport information from one complex…
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high…
Results on $8$-dimensional topological planes are scattered in the literature. It is the aim of the present paper to give a survey of these geometries, in particular of information obtained after the appearance of the treatise Compact…
Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the…
A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and…
Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…
We study complex analytic (possibly singular) projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the…
This article contains a short and entertaining list of unsolved problems in Plane Geometry. Their statement may seem naive and can be understood at an elementary level. But their solutions have refused to appear for forty years in the best…
In this paper we give several methods to construct curves over finite fields with many points and illustrate this with examples of the results.
Call a curve $C \subset \mathbb{P}^2$ defined over $\mathbb{F}_q$ transverse-free if every line over $\mathbb{F}_q$ intersects $C$ at some closed point with multiplicity at least 2. In 2004, Poonen used a notion of density to treat Bertini…
We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an…
We obtain new examples and the complete list of the rational cuspidal plane curves $C$ with at least three cusps, one of which has multiplicity ${\rm deg}\,C - 2$. It occurs that these curves are projectively rigid. We also discuss the…
This paper examines the relationship between the knotting of an embedded surface in $\R^3$ and the knotting of its fold curves, formed by the singular set of projection to a plane. The first result shows that every surface, no matter how…
A new family of maximal curves over a finite field is presented and some of their properties are investigated.
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 dimensions of the graded quotients of the cohomology of a plane curve complement with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed…
We construct for any smooth projective curve of genus $q\ge 2$ with a fixed point free automorphism a nonisotrivial family of curves. Moreover we study the space of modular curves and that of parameters.