Classification of conic bundles on a rational elliptic surface in any characteristic

TL;DR

This paper classifies all possible fibers of conic bundles on rational elliptic surfaces over algebraically closed fields of any characteristic, analyzing their relationship with singular fibers of elliptic fibrations.

Contribution

It provides a comprehensive classification of conic bundle fibers on rational elliptic surfaces in any characteristic using numerical methods.

Findings

Complete classification of conic bundle fibers

Analysis of interaction between singular fibers of elliptic and conic fibrations

Applicable to fields of arbitrary characteristic

Abstract

Let $X$ be a rational elliptic surface with elliptic fibration $\pi:X\to\Bbb{P}^1$ over an algebraically closed field $k$ of any characteristic. Given a conic bundle $\varphi:X\to\Bbb{P}^1$ we use numerical arguments to classify all possible fibers of $\varphi$ and study the interplay between singular fibers of $\pi$ and $\varphi$.