Quadratic centers defining elliptic surfaces

TL;DR

This paper classifies quadratic vector fields with centers that produce genus one algebraic orbits and links each to an associated elliptic surface, providing a comprehensive list and characterization.

Contribution

It provides a complete classification of quadratic centers defining elliptic surfaces and explicitly determines the associated elliptic surfaces for each case.

Findings

List of all quadratic vector fields with genus one centers

Explicit determination of corresponding elliptic surfaces

Classification of these vector fields and surfaces

Abstract

Let $X$ be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one. To each $X$ we associate an elliptic surface (a smooth complex compact surface which is a genus one fibration). We give the list of all such vector fields and determine the corresponding elliptic surfaces.