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

Abstract

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

arXiv comment24 pages, 10 figures