Related papers: Counting higher order tangencies for plane curves

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves…

These notes are intended as an easy-to-read supplement to part of the background material presented in my talks on enumerative geometry. In particular, the numbers $n_3$ and $n_4$ of plane rational cubics through eight points and of plane…

We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is…

We construct a set of points with $\Omega(n^2\log n)$ triples determining an angle $\theta$ whenever $\tan(\theta)$ is algebraic over $\mathbb{Q}$, matching the upper bound of Pach and Sharir. This improves upon the original construction,…

We show that a general plane curve of degree at least 4 is uniquely determined by the full set of its bitangent lines. This problem has an elementary solution for degree at least 5, and the paper is almost entirely devoted to curves of…

A basis of the ideal of the complement of a linear subspace in a projective space over a finite field is given. As an application, the second largest number of points of plane curves of degree $d$ over the finite field of $q$ elements is…

In this review paper we present a stable Lagrangian numerical method for computing plane curves evolution driven by the fourth order geometric equation. The numerical scheme and computational examples are presented.

We use twisted stable maps to compute the number of rational degree d plane curves having prescribed contacts to a smooth plane cubic.

We prove that there are $O(n)$ tangencies among any set of $n$ red and blue planar curves in which every pair of curves intersects at most once and no two curves of the same color intersect. If every pair of curves may intersect more than…

Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,k-3)+3= (n+1)+n+\cdots+(n-k+5)+3$ and $4 \le k\le n-1.$ In this paper we prove that…

We study the order of tangency between two manifolds of same dimension and give that notion three quite different geometric interpretations. Related aspects of the order of tangency, e.g., regular separation exponents, are also discussed.

For every $k>3$, we give a construction of planar point sets with many collinear $k$-tuples and no collinear $(k+1)$-tuples. We show that there are $n_0=n_0(k)$ and $c=c(k)$ such that if $n\geq n_0$, then there exists a set of $n$ points in…

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…

This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further…

Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k…

In the projective plane PG(2,q) over a finite field of order q, a Tallini curve is a plane irreducible (algebraic) curve of (minimum) degree q+2 containing all points of PG(2,q). Such curves were investigated by G. Tallini in 1961, and by…

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and…

The goal of this text is to present the computation by Salmon, in the second half of the XIXth century, of various numbers enumerating planes with a prescribed tangency pattern with a sufficiently general surface $S$ in $\mathbf{P}^3$ (or,…

Computing the topology of an algebraic plane curve $\mathcal{C}$ means to compute a combinatorial graph that is isotopic to $\mathcal{C}$ and thus represents its topology in $\mathbb{R}^2$. We prove that, for a polynomial of degree $n$ with…

We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most…