Related papers: On maps taking lines to plane curves
We consider the parameterization ${\mathbf{f}}=(f_0,f_1,f_2)$ of a plane rational curve $C$ of degree $n$, and we want to study the singularities of $C$ via such parameterization. We do this by using the projection from the rational normal…
A novel family of integrable third order maps is presented. Each map possesses, by construction, a pair of rational invariants and a commuting map from the same class. The 3-dimensional invariant curve is parametrized, in general, by an…
We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense.…
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Finding such a cuboid is equivalent to finding a perfect cuboid with all…
This will is an expository description of quadratic rational maps. Sections 2 through 6 are concerned with the geometry and topology of such maps. Sections 7--10 survey of some topics from the dynamics of quadratic rational maps. There are…
We classify surfaces of general type whose bicanonical map is composed with a rational map of degree 2 onto a rational or ruled surface.
We investigate straight-line drawings of topological graphs that consist of a planar graph plus one edge, also called almost-planar graphs. We present a characterization of such graphs that admit a straight-line drawing. The…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
We present a complete classification of complex plane algebraic curves, equipped with the induced Euclidean, up to global bilipschitz homeomorphism.
Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $\alpha_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right] $ in the Jacobian of $C$. In…
We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the…
It is a classical result that there are $12$ (irreducible) rational cubic curves through $8$ generic points in $\mathbb{P}_{\mathbb{C}}^2$, but little is known about the non-generic cases. The space of $8$-point configurations is…
One describes those double structures on rational normal curves which are defined scheme theoretically by quadratic equations and have linear syzygies, generalizing this way the double line in the plane
For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane…
We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial…
This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal…
We determine all of lines in the moduli space $M$ of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section.
In the nineties, several methods for dealing in a more efficient way with the implicitization of rational parametrizations were explored in the Computer Aided Geometric Design Community. The analysis of the validity of these techniques has…