By Igor Dolgachev, Anatoly Libgober (auth.), Anatoly Libgober, Philip Wagreich (eds.)

ISBN-10: 3540108335

ISBN-13: 9783540108337

**Read or Download Algebraic Geometry: Proceedings of the Midwest Algebraic Geometry Conference, University of Illinois at Chicago Circle, May 2 – 3, 1980 PDF**

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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**

**Sample text**

The [x 0 ..... x m] the two Ix,y] complement = p2m+l [Y] of and of Y0 = if U , then = pm the from the diagonal Lm c ~2m+l [Y0 ..... Y m ] × ~m [x 0 ..... X m , Y 0 ..... y m ] in ... ~ Z l ( X ) idea to a l i n e a r is c o n n e c t e d . m x ]~m V L : X ÷ , by as a linear maps C*-bundle space isomorphically ~m x pm , set 40 and let q situation : X* + X and is s u m m a r i z e d f* : X* + V in the following m v L , ~' the p r o j e c t i o n s . 2. 3. this the ]~ f* linear sections proves U* commutative is pro- (A) .

Spaces finds of these theorems §2 w e Over groups of results variety: and fundamental covering theorem strengthens irreducible linear on c ~m to B e r t i n i - t y p e linear arbitrary results and devoted sections an , which hyperplane the §I pass the by on to com- apply- varieties in question. The form that connectedness if X theorem, is a c o m p l e t e f is a m o r p h i s m such that proved in §3, irreducible : X + 1Dm dimf(X) > m asserts variety, in its and simplest if x pm , then the inverse image f-l(A) 29 of the diagonal case, the that X we × pm homomorphism is present brief, £ c pm locally is d u e one uses a morphism zl(f-l(A)) irreducible to Deligne, a basic f* is c o n n e c t e d .

Ing a b i r a t i o n a l c o r r e s p o n d e n c e between It depends upon construct- pm x ~m and p2m which reduces the a s s e r t i o n for the diagonal to the c o r r e s p o n d i n g statement for a linear space originally Lm c ~ 2 m (B) is due to Deligne [i0, ii], who proved it using the b i r a t i o n a l correspondence. of the c o n n e c t e d n e s s The proof theorem p r e s e n t e d above was given by D e l i g n e [12] 42 in t h e (cf course of extending the In t h e situation of theorem to h i g h e r homotopy groups §9).