Related papers: Singular plane curves: freeness and combinatorics
We consider here square tilings of the plane. By extending the formalism introduced in [3] we build a correspondence between plane maps endowed with an harmonic vector and square tilings satisfying a condition of regularity. In the case of…
We consider links of complex isolated hypersurface singularities in $\mathbb{C}^{n+1}$ and study differentiable maps defined by restricting holomorphic functions to the links. We give an explicit example in which such a restriction gives a…
Recently Fukasawa, Homma and Kim introduced and studied certain projective singular curves over $\mathbb {F}_q$ with many extremal properties. Here we extend their definition to more general non-rational curves.
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…
In the time evolution of fluids, the topologies of fluids can be changed by the creations and annihilations of singular points and by switching combinatorial structures of separatrices. In this paper, to describe the possible generic time…
In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of…
We study obstacle problems governed by two distinct types of diffusion operators involving interacting free boundaries. We obtain a somewhat surprising coupling property, leading to a comprehensive analysis of the free boundary. More…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
In this paper, we introduce the interested reader to homological mirror symmetry. After recalling a little background knowledge, we tackle the simplest cases of homological mirror symmetry: curves of genus zero and one. We close by…
Consider a smooth projective curve and a given embedding into projective space via a sufficiently positive line bundle. We can form the secant variety of $k$-planes through the curve. These are singular varieties, with each secant variety…
There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…
We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We…
This study introduces a new type of general helix called associated helix which is associated to a special surface curve. The basic idea is to determinate the parametric form of an associated helix by means of Darboux frame and surface…
In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum…
The paper introduces a number of new techniques to handle minimal hyersurface singularities. In particular, they allow to extend the obstruction theory for postive scalr curvature to any dimension.
This is a survey of recent progress in several areas of combinatorial algebra. We consider combinatorial problems about free groups, polynomial algebras, free associative and Lie algebras. Our main idea is to study automorphisms and, more…
A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first…
The Hodge spectrum is an important analytic invariant of singularities encoding the Hodge filtration and the monodromy of the Milnor fiber. Explicit formulas exist for only a few cases. In this article the main result is a combinatorial…
This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs $(C,o)$ of complex analytic curves contained in a smooth complex analytic surface $S$. The…
The nature of the initial singularity in spatially compact plane symmetric scalar field cosmologies is investigated. It is shown that this singularity is crushing and velocity dominated and that the Kretschmann scalar diverges uniformly as…