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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Additional resources for Algebraic Geometry: Proceedings of the Midwest Algebraic Geometry Conference, University of Illinois at Chicago Circle, May 2 – 3, 1980

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7. Y x Algebraic (A) In fact, X X is c o n n e c t e d . varieties I_~f X is an closed algebraic X- Z enjoy (irreducible) subset, is an i r r e d u c i b l e is an i r r e d u c i b l e the analytic space, variety, X- Z variety. and Z and property: Z ~ X is c o n n e c t e d . The same is a c l o s e d is t r u e w h e n analytic sub- X 33 space; this follows easily from the c o r r e s p o n d i n g proved in local statement [28, p. i15]. (B) If X is an i r r e d u c i b l e complex v a r i e t y whose univer- sal c o v e r i n g ~ is an irreducible analytic then for any closed analytic subspace space, Z ~ X , ~i (X-Z)--~ ~I(Z) Indeed, by the p r e v i o u s remark To make use of irreducible.

36 It follows for w h i c h from statements for later purposes. w i t h the d e s i r e d consisting that there (A) and properties, the p r o j e c t i o n s finite m o r p h i s m on M set of hold. set disjoint considered ~ x V ÷ ~n × V Y , and trivializes above Choose This suffices TM) V ~ G r a s S n + l ( P m) from G r a s s n + l ( P m) G r a s s d ( P m) U ~ Grassd(~ a Zariski-open M c ~m bundle from a Zariski-open choose spaces quotient is a dense (B) of the t h e o r e m To produce of linear the u n i v e r s a l Then (*) fit t o g e t h e r a divisor B c pn such that over V to form a × V such that the c o m p o s i t i o n X is a t o p o l o g i c a l V surfaces U covering if necessary, S to consist of all corresponding L' versely b points.

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. Other Moishezon b y K. , for the o b s t r u c t i o n that positive theorem meets c a s e of C o r o l l a r y two homology is w e a k l y Tan(X) that L c ~r but not f : X ÷ pm contradicts formulas be homologous the p r o j e c t i o n L discovered showed map , but space Sec(X) not meet since ~ Sec(X) a linear meets a finite this Tan(X) choose or one-to-one, calculation vanishing gular was unramified formal But special ~2n the that L does hand, be o n e - t o - o n e .

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