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An algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
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Isotrivial fibrations over $\mathbb P^1$
There are three important invariants for any relatively minimal fibration $f:\,S \to B$ from a smooth complex projective surface $S$ to a smooth curve $B$: the self-intersection $\omega_{S/B}^2$, the …