Related papers: Computing curves on real rational surfaces

In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to…

We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{CP}^2$ that pass through $3d+1-m$ generic points and that have an $m$-fold singular point. The special case of counting curves with a triple point was…

We give several algorithms addressing computations of intersections of conjugate subgroups.

In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…

We prove the existence of embedded closed constant curvature curves on convex surfaces.

The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is desirable to…

In this paper we present a method for extraction of arcs of the algebraic curves of the higher order. Method is applied on conics, Cartesian ovals, trifocal curves and generalized Weber's curve.

A new family of maximal curves over a finite field is presented and some of their properties are investigated.

We give an algorithm to compute the conductor for curves of genus 2. It is based on the analysis of 3-torsion of the Jacobian for genus 2 curves over 2-adic fields.

We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular,…

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

Let p be a prime and let C be a genus one curve over a number field k representing an element of order dividing p in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of D in the Shafarevich-Tate…

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 present a characterisation of blenders based on mapping properties of certain sets of curves that can be rigorously verified by computer-assisted methods. We develop an algorithm to construct these sets of curves that requires only a…

These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in G\"ottingen, June 2007. The first part gives an overview of the methods. The main new result is the…

We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients…

This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms…

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…