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

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

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