Related papers: Counting real curves with passage/tangency conditi…

If two closed Jordan curves in the plane have precisely one point in common, then it is called a {\em touching point}. All other intersection points are called {\em crossing points}. The main result of this paper is a Crossing Lemma for…

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is…

There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for reducible real quintic curves. The classification of real singular points for irreducible real quintic curves is…

We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination…

Consider l lines in P^2 such that no three lines meet in a point. Let X(l) denote all points of intersections of these l lines. We describe all pairs (d,l) such that generic degree d curve in P^2 contains a X(l).

It is well known that a non-singular real plane projective curve of degree five with five connected components is separating if and only if its ovals are in non-convex position. In this article, this property is set into a different context…

The average value of the number of intersection points of two plane curves given by trigonometrical polynomial maps of degree $N$ and bounded $L_2$- or $W_2^r$- norms is calculated

The paper gives topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of…

We prove that the moduli space of contact stable maps to $\mathbb{P}^{2n+1}$ of degree $d$ admits a stratification parameterized by graphs. We use it to determine the number of irreducible rational contact curves in $\mathbb{P}^{2n+1}$ with…

Let $P$ be a set of $n$ points in real projective $d$-space, not all contained in a hyperplane, such that any $d$ points span a hyperplane. An ordinary hyperplane of $P$ is a hyperplane containing exactly $d$ points of $P$. We show that if…

On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb P^n$ itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number $t$ of general points or…

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…

Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower…

We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic $M$-curves realize many isotopy types. The strategy we…

We discuss the problem of whether a given problem in enumerative geometry can have all of its solutions be real. In particular, we describe an approach to problems of this type, and show how this can be used to show some enumerative…

We obtain a formula for the degrees of the varieties parameterizing complex algebraic curves of any divisor class and genus on P^2_6, the projective plane blown-up at 6 generic points. Moreover, the formula computes the degrees of the…

Let $G$ be a subgroup of the three dimensional projective group $\mathrm{PGL}(3,q)$ defined over a finite field $\mathbb{F}_q$ of order $q$, viewed as a subgroup of $\mathrm{PGL}(3,K)$ where $K$ is an algebraic closure of $\mathbb{F}_q$.…

We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general…

This is an exposition of a class of problems and results on the number of integral points close to plane curves. We give a detailed proof of a theorem of Huxley and Sargos, following the account of Bordell\`es. Along the way we correct an…