Related papers: Closed Cycloids in a Normed Plane

We prove the regularity conjecture, namely Eisenbud-Goto conjecture, for a normal surface with rational, Gorenstein elliptic and log canonical singularities. Along the way, we bound the regularity for a dimension zero scheme by its Loewy…

Let $\mathbf{R}_d$ be the space of stable sheaves $F$ which satisfy the Hilbert polynomial $\chi(F(m))=dm+1$ and are supported on rational curves in the projective plane $\mathbb{P}^2$. Then $\mathbf{R}_1$ (resp. $\mathbf{R}_2$) is…

The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane…

If a curve in R^3 is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a 2x2 matrix…

This paper focuses on curves and surfaces of constant width, with some additional results about general ovals. We emphasize the use of Fourier series to derive properties, some of which are known. Amongst other results, we show that the…

The Tait-Kneser theorem states that the osculating circles of a plane curve with monotonic curvature are pairwise disjoint and nested. We discuss this theorem and a number of its variations.

I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve.…

The skewer of a pair of skew lines in space is their common perpendicular. To configuration theorems of plane projective geometry involving points and lines (such as Pappus or Desargues) there correspond configuration theorems in space:…

We show that the universal plane curve M of fixed degree d > 2 can be seen as a closed subvariety in a certain Simpson moduli space of 1-dimensional sheaves on a projective plane contained in the stable locus. The universal singular locus…

We establish a Poincar\'e-Dulac theorem for sequences (G_n)_n of holomorphic contractions whose differentials d_0 G_n split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of…

By using a generalization of Sturm-Liouville problems in discrete spaces, a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters, which generalizes all classical discrete symmetric orthogonal…

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational…

We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and…

We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of…

We show that $\mathbb{C}^2$ contains pairs of properly embedded, smooth complex curves that are isotopic through homeomorphisms but not diffeomorphisms of $\mathbb{C}^2$. The construction is based on realizing corks as branched covers of…

In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in $\mathbb{R}^3$. We show this self-similar solutions are the…

Let $f: X \to Y$ be a regular covering of a surface $Y$ of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group $G$. We give necessary and sufficient conditions for an integral homology class on $X$ to…

Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable…

If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex…

We establish necessary conditions for a regular curve to lie on a circular cylinder in terms of its curvature $\kappa$ and torsion $\tau$. By identifying a fundamental function $\psi = \sin^2 \alpha$, representing the squared sine of the…