Related papers: Unexpected hypersurfaces and their consequences: A…
We survey various Alexander-type invariants of plane curve complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to complex plane curves. Also included are some new…
Quivers, gauge theories and singular geometries are of great interest in both mathematics and physics. In this note, we collect a few open questions which have arisen in various recent works at the intersection between gauge theories,…
In this note we introduce the notion of $(b,d)$-geprofi sets and study their basic properties. These are sets of $bd$ points in $\mathbb{P}^4$ whose projection from a general point to a hyperplane is a full intersection, i.e., the…
The geproci property is a recent development in the world of geometry. We call a set of points $Z\subseteq\mathbb{P}_k^3$ an $(a,b)$-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point $P$…
In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also,…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
Very few examples of obstructed equsingular families of curves on surfaces other than the projective plane are known. Combining results from Westenberger and Hirano with an idea from math.AG/9802009 we give in the present paper series of…
We study the probability for a random line to intersect a given plane curve, defined over a finite field, in a given number of points defined over the same field. In particular, we focus on the limits of these probabilities under successive…
Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two…
Extracting planes from a 3D scene is useful for downstream tasks in robotics and augmented reality. In this paper we tackle the problem of estimating the planar surfaces in a scene from posed images. Our first finding is that a surprisingly…
Conics and Cartesian ovals are very important curves in various fields of science. Also aspheric curves based on conics are useful in optics. Superconic curves recently suggested by A. Greynolds are extensions of both conics and Cartesian…
The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to…
We study the principal curvatures of properly embedded constant mean curvature hypersurfaces in the Anti-de Sitter space $\mathbb{H}^{n,1}$. We generalize the notion of convex hull and give an upper bound on the principal curvatures which…
We study the integer sequence v_n of numbers of lines in hypersurfaces of degree 2n-3 of P^n, n>1. We prove a number of congruence properties of these numbers of several different types. Furthermore, the asymptotics of the v_n are described…
Planar curves with periodically varying curvature arise in the natural sciences as the result of a wide variety of periodic processes. The total curvature of a periodic arc in such curves constrains their symmetry. It is shown how the total…
This article consists of two parts. The first part is a survey on the normal reduction numbers of normal surface singularities. It includes results on elliptic singularities, cone-like singularities and homogeneous hypersurface…
We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite…
Given a point set, mostly a grid in our case, we seek upper and lower bounds on the number of curves that are needed to cover the point set. We say a curve covers a point if the curve passes through the point. We consider such coverings by…
In this short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the…
These are lecture notes from a mini-course taught at Winterbraids XIII (Montpellier, 2024). The main character of these notes are curves in the complex projective plane, viewed from a topological perspective.