By Igor Dolgachev, Anatoly Libgober (auth.), Anatoly Libgober, Philip Wagreich (eds.)
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Extra info for Algebraic Geometry: Proceedings of the Midwest Algebraic Geometry Conference, University of Illinois at Chicago Circle, May 2 – 3, 1980
Contrary T h e n we m a y m s 2n - 1 not to t h e < 2n codimension fied. : Sec(X) cases and of Corollary D. 5 h a d b e e n (n=2), was a n d J. discovered proved Harris before (n=2,3). independently  b y B. For non-sin- b y F. 3(B) follows . Higher Coverings coverings be a complete variety  subseq- case.  for n o n - s i n -  w h e n X is a local and by H a r t s h o r n e in c h a r a c t e r i s t i c suggestions homotopy theorem and p of Deligne analogues of P r o j e c t i v e the c o n n e c t e d n e s s of b r a n c h e d and L a r s e n zero, set- simple-connectiv- to the a b s t r a c t by Ogus of the In the a l g e b r a i c (algebraic) by Barth F r o m the §9) that any h y p e r s u r f a c e and G r o t h e n d i e c k in c h a r a c t e r i s t i c Branched X degenerate consequence connected.
Is used to study the of p r o j e c t i v e space. of d i m e n s i o n n , and let f : X ÷ ~n be a finite morphism. e. 1. ef(x) The proof theorem of p r o j e c t i v e () (A) >- m i n ( d , will generalizes yield There space exists ramification U c X £+I image in is the pn a stronger loci set of , then £ < min(d Proof n = 1 If L c pn * the n ~ 2 , the In t h e fact that every ramify. at l e a s t one point x e X at are i ef(x) closed Namely, R£ = U n A x the sets > £} algebraic (£+l)-tuples consider subsets of d i s t i n c t We will of points show that X : for if with the same in f a c t , X) ~ £ - i, n) of T h e o r e m being must statement.
1. For the diagram ~l(X) irreducible, homomorphism is is of T h e o r e m ) ~ l ( X × Y*) ~l(f-l(Y)) and let subvariety. T h e n one has the c o m m u t a t i v e X is c o n n e c t e d . then variety, irreducible; Zl(X × p m Since and if is c o n n e c t e d . be the n o r m a l i z a t i o n i n d u c e d map. ×i~m X r ) - - - ~ Z l ( X 1 × . . 1(B) surjective. surjective, and the applies to But the Corollary • (I) the C o r o l l a r y Hironaka when f and Matsumura is s u r j e c t i v e .  h a d p r o v e d The t h e o r e m assertion (A) of s t a t e d by B a r t h in 45  was zation Rossi the special of a s u b v a r i e t y of the c o r o l l a r y of pm The conjecture in varieties, and then f-l(y) dimf(X) and the  if f : X ÷ Z if that Y Z-~ Z would be c o n n e c t e d is a m o r p h i s m with Hansen (cf.