Related papers: Unexpected hypersurfaces and their consequences: A…
In this paper we consider two notions that have been discovered and rediscovered by geometers and analysts since 1917 up to the present day: Radon curves and antinorms. A Radon curve is a special kind of centrally symmetric closed convex…
In this note we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we…
In the paper we provide a new method of proving the existence of a hypersurface of degree $d$ in $\mathbb{P}^n$, with a general point of multiplicity $m$ and vanishing at a given set of points $Z$, by looking at weak combinatorics of a set…
We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the…
A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. In this paper, we study $\mathbb{A}^1$-curves on the complement of a rational strange curve of degree $p$ in…
Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an…
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…
We give a complete classification of polynomial models for smooth real hypersurfaces of finite Catlin multitype in $\mathbb C^3$, which admit nonlinear infinitesimal CR automorphisms. As a consequence, we obtain a sharp 1-jet determination…
Given two curves in $\PP^3$, either implicitly or by a parameterization, we want to check if they intersect. For that purpose, we present and further develop generalized resultant techniques. Our aim is to provide a closed formula in the…
This paper provides an overview of selected results and open problems in the theory of hyperplane arrangements, with an emphasis on computations and examples. We give an introduction to many of the essential tools used in the area, such as…
This is a survey describing recents developments in enumerative geometry of curves on projective varieties. Various methods to arrive at results such as Kontsevich's formula for plane rational curves, or Caporaso-Harris's formula for plane…
The purpose of this note is to present and study a new series of the so-called unexpected curves. They enjoy a surprising property to the effect that their degree grows to infinity, whereas the multiplicity at a general fat point remains…
Most major discoveries in astronomy are unplanned, and result from surveying the Universe in a new way, rather than by testing a hypothesis or conducting an investigation with planned outcomes. For example, of the 10 greatest discoveries…
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…
A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…
We first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves…
Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint.…
In 1983, Nochka proved a conjecture of Cartan on defects of holomorphic curves in CP^n relative to a possibly degenerate set of hyperplanes. In this paper, we generalize the Nochka's theorem to the case of curves in a complex projective…
We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and…
We give an overview of results on irregular complex surfaces of general type, discussing in particular the distribution of the numerical invariants self-intersection of a canonical divisor and holomorphic Euler characteristic for the…