Fundamental groups of algebraic curves

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In the previous chapter the Riemann Existence Theorem created a link between the category of compact connected Riemann surfaces and that of finite extensions of C ( t ). This hints at a possibility of developing a theory of the fundamental group in a purely algebraic way. We shall now present such a theory for curves over an arbitrary perfect base field, using a modest amount of algebraic geometry. Over the complex numbers the results will be equivalent to those of the previous chapter, but a new and extremely important feature over an arbitrary base field k will be the existence of a canonical quotient of the algebraic fundamental group isomorphic to the absolute Galois group of k . In fact, over a subfield of C we shall obtain an extension of the absolute Galois group of the base field by the profinite completion of the topological fundamental group of the corresponding Riemann surface over C . This interplay between algebra and topology is a source for many powerful results in recent research. Among these we shall discuss applications to the inverse Galois problem, Belyi's theorem on covers of the projective line minus three points and some advanced results on ‘anabelian geometry’ of curves. Reading this chapter requires no previous acquaintance with algebraic geometry. We shall, however, use some standard results from commutative algebra that we summarize in the first section. The next three sections contain foundational material, and the discussion of the fundamental group itself begins in Section 4.5.