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Jan 14, 2017 at 0:11 comment added Jason Starr @IMeasy. Yes, I think the argument works very generally. I learned after I wrote this answer that Pardini has a lovely article on Abelian covers. I am certain the answer above follows from her article. Probably what you ask also follows from Pardini's article.
Jan 13, 2017 at 10:05 comment added IMeasy Dear Jason, is it correct to say that your argument works for projective spaces (or even smooth quadrics) of any dimension, if one assumes that singularities of the ramification are good enough (say, double points)? Of course $U$ would just be an open set whose complement is at least codimension two.
S Dec 26, 2013 at 14:12 history answered Jason Starr CC BY-SA 3.0
S Dec 26, 2013 at 14:12 history made wiki Post Made Community Wiki by Jason Starr