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

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3) an a r b i t - covering f : X ÷ ~n This is s h o w n by a n e x a m p l e f' : X' ÷ ~ n of d e g r e e be the variety in t h e same of d e g r e e also gives obtained fibre d of , but L c pn (compare Theorem Hansen [32] have due by f' Then ~I(X) an e x a m p l e fails and and generalizes This + Zl(X) (unpublished) normal in X ~l(f-l(L)) hyperplane ~I(X) irreducible a branched f : X ÷ ~n the h o m o m o r p h i s m for a generic in a covering and if . Theorem then every of the b r a n c h space, R1 covering ~ £ connected.

35] w h e n in his proof §7). local of view, at least Abhyankar (cf. surface hyperplane of d i m e n s i o n role varieties. that a n o n - s i n g u l a r Lefschetz a crucial for of this Space. 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 ([20]) (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.

E. 1(B), is s i m i l a r (*) that × ~I(X) by the d i a g o n a l 5y T h e o r e m argument a trivial + ~I(X) I f*×f* - X~_~ X X X e e Im(f,) The commutative is the h o m 0 m o r p h i s m 6 : X = X ×~m : ~l(X*) >~l(X*× I where f, ×]pm X*) ZI(X) for be a c l o s e d is u n r a m i f i e d , a section points. F = f × f × f : X × X × X ÷ 1D4 × ~ 4 × i~4 . has no n o n - t r i v i a l Suppose Y' + X f - ~ m of p i n c h for X ¢ pm be an i r r e d u c i b l e Y' first F -I(A) is s i m p l y (A) Proof.

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