# Why is a smooth algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ of degree 5 simply connected?

The title says it all.

In fact, i am only trying to prove that if $S$ is an irreducible smooth algebraic surface of degree 5 in $\mathbb{P}_{\mathbb{C}}^3$ (hence a four dimensional manifold over the reals), its first singular cohomology group $H_1(S ; \mathbb{Z})$ contains no two torsion.

I know only that by the short exact sequence of the sheaf of ideals and hodge theory that this group has no free part. But for the torsion part i have no idea.

Any help would be really appreciated, i would prefer hints over a complete answer!

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## 1 Answer

Hint: this follows from the Lefschetz Hyperplane Theorem using the Veronese (or "d-uple") Embedding. See e.g. here for the details, if you like.

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Awesome, thanks a lot! I did not know about that theorem (and the elegant way to apply it) before.. –  Joachim Dec 19 '12 at 21:21
@Joachim: You're welcome. The Lefschetz Hyperplane Theorem is indeed a powerful and useful result in the topology of complex varieties. At some point you'll probably want to look up the "Hard Lefschetz Theorem" as well. –  Pete L. Clark Dec 19 '12 at 23:20