Related papers: On the self-intersection number of the nonsingular…
In this paper, we consider rational cuspidal plane curves having exactly two cusps whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal…
In this note, we consider rational cuspidal plane curves having exactly one cusp whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal embedded…
In this article we give an upper bound for the number of cusps on a cuspidal curve on a Hirzebruch surface. We adapt the results that have been found for a similar question asked for cuspidal curves on the projective plane, and restate the…
In this paper we consider smooth affine elliptic plane curves having one place at infinity. We identify them with elliptic projective plane curves having only one cusp as their singular points and meeting with the line at infinity only at…
We obtain new examples and the complete list of the rational cuspidal plane curves $C$ with at least three cusps, one of which has multiplicity ${\rm deg}\,C - 2$. It occurs that these curves are projectively rigid. We also discuss the…
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 show that the 1-planar slope number of 3-connected cubic 1-planar graphs is at most 4 when edges are drawn as polygonal curves with at most 1 bend each. This bound is obtained by drawings whose vertex and crossing resolution is at least…
The purpose of this article is to shed light on the question of how many and what kind of cusps a rational cuspidal curve on a Hirzebruch surface can have. We use birational transformations to construct rational cuspidal curves with four…
In this paper we obtain a formula for the number of rational degree d curves in $\mathbb{P}^3$ having a cusp, whose image lies in a $\mathbb{P}^2$ and that passes through $r$ lines and $s$ points (where $r + 2s = 3d + 1$). This problem can…
We give an upper bound for the number of cusps of a plane affine or projective curve via its first Betti number.
We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.
Let C be a smooth cubic curve in the complex projective plane. We show that for every positive integer k, there are only finite number of rational curves of degree k each intersects the cubic C at exactly one point. The number of such…
A symplectic rational cuspidal curve with positive self-intersection number admits a concave neighborhood, and thus a corresponding contact manifold on the boundary. In this article, we study symplectic fillings of such contact manifolds,…
We give a practical formula for counting irreducible nodal genus-three plane curves that a fixed generic complex structure on the normalization. As an intermediate step, we enumerate rational plane curves that have a $(3,4)$-cusp.
Given two general rational curves of the same degree in two projective spaces, one can ask whether there exists a third rational curve of the same degree that projects to both of them. We show that, under suitable assumptions on the degree…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
We completely classify all plane curves of degree at most 30 with a unique cuspidal (locally unibranch) singular point and rational normalization in terms of the Newton pairs parameterizing the cusp. We distinguish between prime and…
Given two distinct reduced, irreducible curves of given degrees, contained in projective space but whose union is not contained in a hyperplane, what is the largest number of points of intersection they can have? When the projective space…
We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity 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…