Related papers: Computing curves on real rational surfaces
We prove that the Euler-Chow series for ruled surfaces and scrolls is rational by means of an explicit computation.
We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative…
In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and…
We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler…
A new, simple method to approach enumerative questions about rational curves on rational surfaces is described. Applications include a short proof of Kontsevich's formula for plane curves and a the solution of the analogous problem for the…
We present an algorithm for detecting basepoints of linear series of curves in the plane. Moreover, we give an algorithm for constructing a linear series of curves in the plane for given basepoints. The underlying method of these algorithms…
A simple and efficient algorithm to numerically compute the genus of surfaces of three-dimensional objects using the Euler characteristic formula is presented. The algorithm applies to objects obtained by thresholding a scalar field in a…
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the…
An algorithm is given to compute a normal form for hyperelliptic curves. The elliptic case has been treated in a previous paper. In this paper the hyperelliptic case is treated.
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…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
We prove that there is an algorithm to determine if a given finite graph is an induced subgraph of a given curve graph.
In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential…
In this paper, we first summarize the existing algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. From these previous results, we derive a method that allows to easily…
This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify…
An expository description of smooth cubic curves in the real or complex projective plane.
We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.
We define a signed count of real rational pseudo-holomorphic curves appearing in a one-parameter family of real Spin symplectic K3 surfaces. We show that this count is an invariant of the deformation class of the family. In the case of a…
We describe a practical algorithm to compute the (oriented) genus of a graph, give results of the program implementing this algorithm, and compare the performance to existing algorithms. The aim of this algorithm is to be fast enough for…
This is a survey describing recents developments in enumerative geometry of curves on projective varieties. Various methods to arrive at results such as Kontsevich's formula for plane rational curves, or Caporaso-Harris's formula for plane…