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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…
The goal of this paper is to explore the genus and degree of the Fano scheme of linear subspaces on a complete intersection in a complex projective space. Firstly, suppose that the expected dimension of the Fano scheme is one, we prove a…
We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.
To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
Let $X$ be either a general hypersurface of degree $n+1$ in $\mathbb P^n$ or a general $(2,n)$ complete intersection in $\mathbb P^{n+1}, n\geq 4$. We construct balanced rational curves on $X$ of all high enough degrees. If $n=3$ or $g=1$,…
Let $X$ be an irreducible projective variety of dimension $n$ in a projective space and let $x$ be a point of $X$. Denote by ${\rm Curves}_d(X,x)$ the space of curves of degree $d$ lying on $X$ and passing through $x$. We will show that the…
On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete…
We study geometric properties of linear strata of uni-singular curves. The singularities of closures of the strata are resolved and the resolutions are represent as projective bundles. This enables to study their geometry. In particular we…
We get sharp degree bound for generic smoothness and connectedness of the space of conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on Hypersurfaces.
A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to…
We consider the relations between different measures of complexity for free homotopy classes of curves on a surface $\Sigma$, including the minimum number of self-intersections, the minimum length of the words representing them in a…
We give an effective iterative characterization of the classes of (smooth, rational) (-1)-curves on the blowup of the projective plane at general points. Such classes are characterized as having self-intersection -1, arithmetic genus 0, and…
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…
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface…
We consider smooth surfaces $S \subset \Pq$ containing a plane curve $P$ and prove some general result concerning the linear system $|H-P|$. We then look at regular surfaces lying on hypersurfaces of degree $s$ having a plane of…
Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer…
We classify closed curves on a once-punctured torus with a single self-intersection from a combinatorial perspective. We determine the number of closed curves with given word-length and with zero, one, and arbitrary self-intersections.
In this paper, the general formulation for inextensible flows of curves on oriented surface in $\mathbb{R}^3 $ is investigated. The necessary and sufficient conditions for inextensible curve flow lying an oriented surface are expressed as a…
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…