Parameter Spaces for Curves on Surfaces and Enumeration of Rational Curves
نویسنده
چکیده
Contents 1. Introduction 2 1.1. The general strategy: the cross-ratio method 2 1.2. Two sample calculations 7 1.3. Notation and Terminology 14 1.4. Summary of results 15 2. Degenerations of rational curves 17 2.1. The basic setup 17 2.2. The main results from deformation theory 20 2.3. The geometry of the Severi varieties 21 2.4. Singularities of the total space 37 3. Formulas 56 3.
منابع مشابه
TENSION QUARTIC TRIGONOMETRIC BÉZIER CURVES PRESERVING INTERPOLATION CURVES SHAPE
In this paper simple quartic trigonometric polynomial blending functions, with a tensionparameter, are presented. These type of functions are useful for constructing trigonometricB´ezier curves and surfaces, they can be applied to construct continuous shape preservinginterpolation spline curves with shape parameters. To better visualize objects and graphics atension parameter is included. In th...
متن کاملEnumeration of One-Nodal Rational Curves in Projective Spaces
We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. The formula involves intersections of tautological classes on moduli spaces of stable rational maps. We combine the methods and results from three different papers.
متن کاملA Tropical Approach to Enumerative Geometry
A detailed algebraic-geometric background is presented for the tropical approach to enumeration of singular curves on toric surfaces, which consists of reducing the enumeration of algebraic curves to that of non-Archimedean amoebas, the images of algebraic curves by a real-valued non-Archimedean valuation. This idea was proposed by Kontsevich and recently realized by Mikhalkin, who enumerated t...
متن کاملA Class of Quasi-Quartic Trigonometric BÉZier Curves and Surfaces
A new kind of quasi-quartic trigonometric polynomial base functions with a shape parameter λ over the space Ω=span {1, sint, cost, sint2t, cos2t} is presented, and the corresponding quasi-quartic trigonometric Bézier curves and surfaces are defined by the introduced base functions. The quasi-quartic trigonometric Bézier curves inherit most of properties similar to those of quartic Bézier curves...
متن کاملRational Points on Atkin-Lehner Quotients of Shimura Curves
We study three families of Atkin-Lehner quotients of quaternionic Shimura curves: X, X 0 (N), and X D+ 1 (N), which serve as moduli spaces of abelian surfaces with potential quaternionic multiplication (PQM) and level N structure. The arithmetic geometry of these curves is similar to, but even richer than, that of the classical modular curves. Two important differences are the existence of a no...
متن کامل