Consider $X=\mathbb{CP}^2-\bigcup C_i$ where $C_i$ are Riemann surfaces intersecting generically.
How to compute the fundamental group of this space and what is the universal cover?
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5$\begingroup$ This case completely different from what happens in one dim. It's a theorem of Fulton that the fundamental group is $\mathbb{Z}/\deg P$. So $\tilde X$ would be complement of a finite cyclic cover of the plane branched over the Riemann surface. $\endgroup$– Donu ArapuraCommented Sep 17, 2023 at 16:44
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2$\begingroup$ @DonuArapura: wouldn't the fundamental group be $F \oplus \mathbb{Z}/D\mathbb{Z}$, where $F$ is free Abelian of rank $\#$components$-1$ and $D = \gcd\{d_i\}$? $\endgroup$– Marco GollaCommented Sep 17, 2023 at 19:29
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4$\begingroup$ Isn't the result of @DonuArapura's first comment due to Zariski already? At any rate, that answers the question as currently stated. If the OP wants to ask something else, I suggest editing the question (since no answer has been posted yet — otherwise write out a new question). $\endgroup$– R. van Dobben de BruynCommented Sep 17, 2023 at 22:10
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3$\begingroup$ @R.vanDobbendeBruyn: the history is a bit complicated. Zariski proved it based on Severi's theorem of the irreducibility the strata of the nodal variety (I think this is what's called), whose proof had a gap (later filled by Harris). In the meantime, Deligne and Fulton proved that Zariski's conjecture independently of Severi's "proof". (I've learnt this from Cogolludo's notes on braid monodromy, in Ann. Math. B. Pascal, 2011.) $\endgroup$– Marco GollaCommented Sep 18, 2023 at 8:13
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2$\begingroup$ @MarcoGolla Yes, I also learned the history this way (I believe from Joe Harris). $\endgroup$– Jason StarrCommented Sep 18, 2023 at 11:20
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1 Answer
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(For the record, I am summarizing the comments here.) Deligne and Fulton have shown that fundamental group of the complement of a nodal curve in $\mathbb{C}\mathbb{P}^2$ is abelian. It follows easily that in the case of smooth curve $C$ of degree $d$, the fundamental group is $\mathbb{Z}/d$. Therefore the universal cover in this case is the complement of the branch curve in the $d$-sheeted cyclic cover of the plane branched over $C$. See above comments for further historical remarks.
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$\begingroup$ You can also do an irreducible nodal curve, and then you get the complement of a plumbing where you have a bunch of pairwise disjoint (-2)-curves, each intersecting the lift of the $C$ transversally twice, in a degree-$d$ hypersurface in $\mathbb{CP}^3$. $\endgroup$ Commented Sep 19, 2023 at 20:41