Algebraic Geometry: Proceedings of the Midwest Algebraic by Igor Dolgachev, Anatoly Libgober (auth.), Anatoly Libgober,

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

Example text

Assume in 41 addition that X is complete. Then nl(f-l(A)) Proof. Choose a n e i g h b o r h o o d is a d e f o r m a t i o n retract of neighborhood x c f-l(A) U V ~I(X) of V f-l(A) Since of the d i a g o n a l A in f with X such that is proper, f-l(u) f-l(£) there exists a c V Fix , and c o n s i d e r the h o m o m o r p h i s m s nl(f-l(£) ,x) ~ Zl(V,x) \Y\ ~i (f-i (U) ,x) induced by inclusions. The top h o r i z o n t a l map is an isomorphism, while the bottom is surjective by Remark follows.

Suppose d i m Tan(X) Projection ( 2n from such that L gives Since L On t h e o t h e r (i) variety proved being of The expected of t h e s e equivalent. cannot Since connectedness remarkable X unramified, cycles B Johnson to p r o j e c t i o n s varieties. ~2n g r e w o u t of the a t t e m p t classes followed is variety that be one-to-one. to e x t e n d A , the on a projective it m u s t can- is a sub- equivalence) it unramif [38]. 4. 5 w h e n generalized algebraic is w e a k l y Proposition Johnson Tan(X) , with f Sec(X) of if The Johnson's result.

The of maps m > n mathematician how it normality. Roberts to This and we Conjecture L. Zak tangent a proof section to a of is d e v o t e d independently suggested results, which F. spaces leads in refer indi- covering. Hartshorne's linear are notions a branched and arguments detailed it of Soviet J. his when different degrees linear finite Zak's the J. letter. reader to arguments. e. X not hyperplane. 1. 4. 4 X that is if normal (and in irreducible). c ~m* in the of follows. We = m - the be the dual variety the set two corollaries P + X* cases the of of hyperplanes P The have ~ L} as dual projection.

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