Timeline for Uniformization of $\mathbb{CP}^2-\bigcup C_i$, where $C_i$ are Riemann surfaces intersecting generically
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2023 at 1:59 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 3 characters in body; edited title
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| Sep 19, 2023 at 18:02 | history | edited | YCor | CC BY-SA 4.0 |
fixed title for readibility
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| Sep 19, 2023 at 14:36 | answer | added | Donu Arapura | timeline score: 4 | |
| Sep 18, 2023 at 21:48 | history | edited | 0x11111 | CC BY-SA 4.0 |
deleted 303 characters in body; edited title
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| Sep 18, 2023 at 19:24 | history | edited | Michael Hardy | CC BY-SA 4.0 |
fixing the conspicuous typographical error
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| Sep 18, 2023 at 11:20 | comment | added | Jason Starr | @MarcoGolla Yes, I also learned the history this way (I believe from Joe Harris). | |
| Sep 18, 2023 at 8:13 | comment | added | Marco Golla | @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.) | |
| Sep 17, 2023 at 22:10 | comment | added | R. van Dobben de Bruyn | 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). | |
| Sep 17, 2023 at 20:43 | comment | added | Donu Arapura | OK, Marco and Jason are right. Leaving my comment from reference. | |
| Sep 17, 2023 at 19:45 | comment | added | Sean Sanford | If you suspect it's well known, maybe add the tag reference request? | |
| Sep 17, 2023 at 19:31 | comment | added | Jason Starr | @DonuArapura The fundamental group of the complement of three transverse lines is a free Abelian group of rank 2. The Deligne - Fulton Theorem (= Zariski’s Conjecture) is that the complement of an at-worst-nodal plane curve has Abelian fundamental group. | |
| Sep 17, 2023 at 19:29 | comment | added | Marco Golla | @DonuArapura: wouldn't the fundamental group be $F \oplus \mathbb{Z}/D\mathbb{Z}$, where $F$ is free Abelian of rank $\#$components$-1$ and $D = \gcd\{d_i\}$? | |
| Sep 17, 2023 at 18:43 | comment | added | Donu Arapura | Yes, Fulton's theorem would apply to this case also. The fundamental group would be the direct sum $\oplus \mathbb{Z}/d_i$, where $d_i$ are degrees of the components. | |
| Sep 17, 2023 at 17:00 | comment | added | 0x11111 | @DonuArapura Is there analogy of Fulton's theorem where we have several Riemann surfaces intersecting generically? ( I know it is a different question, but this is what actually interests me). | |
| Sep 17, 2023 at 16:44 | comment | added | Donu Arapura | 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. | |
| Sep 17, 2023 at 16:09 | history | asked | 0x11111 | CC BY-SA 4.0 |