Related papers: Self-intersection Number of Negative Curves on Fer…
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at…
The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of…
We classify completely the intersections of the Hermitian curve with parabolas in the affine plane. To obtain our results we employ well-known algebraic methods for finite fields and geometric properties of the curve automorphisms. In…
In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is…
The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a…
We describe each multiple curve on the orientable surface of genus-$g$ with $n$ punctures and one boundary component by using this multiple curve's geometric intersection number with the embedded curves in this surface.
We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial…
We investigate arcs on a pair of pants and present an algorithm to compute the self-intersection number of an arc. Additionally, we establish bounds for the self-intersection number in terms of the word length. We also prove that the…
We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…
The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the…
This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other…
The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between dimensions of its two biggest strata is greater than $1$. The set Sigma of…
In this paper, we consider rational cuspidal plane curves having at least three cusps. We give an upper bound of the self-intersection number of the proper transforms of such curves via the minimal embedded resolution of the cusps. For a…
Previously, numerical evidence was presented of a self-intersecting Bezier curve having the unknot for its control polygon. This numerical demonstration resolved open questions in scientific visualization, but did not provide a formal proof…
In this work we study neighborhoods of curves in surfaces with positive self-intersection that can be embeeded as a germ of neighborhood of a curve on the projective plane.
We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in…
In this paper, we introduce numerical cohomology for arithmetic surfaces, which leads to an absolute version of arithmetic Riemann-Roch formula. As an application, we derive an upper bound for the self-intersection number of relative…
The \emph{canonical degree} of a curve $C$ on a surface $X$ is $K_X\cdot C$. Our main result, is that on a surface of general type there are only finitely many curves with negative self--intersection and sufficiently large canonical degree.…
We study the bounded negativity conjecture for non-quaternionic Hilbert modular surfaces and give an explicit bound for the special case of Hirzebruch-Zagier curves on Hilbert modular surfaces.
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…