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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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Confusion with the genus of a curve $Y$ in a ruled surface $X\to C$ such that $Y\to C$ is in...
This was originally posted on MSE, but after a fair amount of time and a bounty it got no response. Unfortunately I have not yet resolved my doubts.
There's a fair bit of setup here, but you don't ne …