Related papers: Maximizing curves viewed as free curves
Given an ordered sequence of $N$-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of $N$ simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the…
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…
In this paper we solve three open problems on maximal curves with Frobenius dimension 3. In particular, we prove the existence of a maximal curve with order sequence (0,1,3,q).
We determine the maximal number of smooth rational degree d curves on a complex K3-surface of degree 2n provided n is sufficiently large as compared to d>1. We obtain precise characterization of configurations of rational degree d curves…
For prime degree hypersurfaces of dimension at least 3, Mori asked if every smooth proper limit is still a hypersurface. Interestingly in dimensions 1 and 2, this is not the case. For example, Griffin constructed explicit families of…
It is well known that among all closed bounded curves in the plane with the given perimeter, the circle encloses the maximum area. There are many proofs in the literature. In this article we have given a new proof using some ideas of Demar.
In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
We study a problem of minimal surfaces with free boundary written in the form of a non convex minimization problem. Our aim is to characterize optimal solutions by finding a suitable calibration field. A natural upper bound of the infimum…
We consider the $L^p$ mapping properties of maximal averages associated to families of curves, and thickened curves, in the plane. These include the (planar) Kakeya maximal function, the circular maximal functions of Wolff and Bourgain, and…
The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits…
This short, expository note proves the existence of the maximal quotient of a variety by free rational curves.
Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the…
In this paper infinite families of examples of irreducible free and nearly free curves in the complex projective plane which are not rational curves and whose local singularites can have an arbitrary number of branches are given. All these…
Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…
We study the Maximum Happy Vertices and Maximum Happy Edges problems. The former problem is a variant of clusterization, where some vertices have already been assigned to clusters. The second problem gives a natural generalization of…
We consider a maximization problem for eigenvalues of the Laplace-Beltrami operator on surfaces of revolution in $\mathbb{R}^3$ with two prescribed boundary components. For every $j$, we show that there is a surface $\Sigma_j$ which…
In the present paper, we focus on a weighted version of the Bounded Negativity Conjecture which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are…
It is proven that for any topological or analytical types of isolated singular points of plane curves, there exists a non-real irreducible plane algebraic curve of degree $d$ which goes through $d^2$ real distinct points and has imaginary…
In the present paper, we study arrangements of smooth plane conics having only nodes and tacnodes as the singularities. We provide an interesting estimation on the number of nodes and tacnodes that depends only on a linear function of the…
We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by A. A. du Plessis and C.T.C. Wall, can be improved when the curve is a line arrangement.