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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6]). e. 5 example, equation 2n is u n r a m i f i e d , (cf. [27, As since by is a c o n n e c t e d is o n e - t o - o n e . 3. e. (B) 2n > m X z~ig(x) provides (B) This follows from (irreducible) subvariety of , then finite ~tale CoVerings, = 1 connected. that p : Y + X component of p is a c o n n e c t e d of Y and h e n c e The Y' ~ X etale covering. 1. , so it is t r i v i a l . Let f : X* ÷ X The homomorphism be the n o r m a l i z a t i o n of X We c l a i m that (*) To see this, consider Zl(X* the 6, top h o r i z o n t a l Im(f,× f,) if 6, map remainder Specifically, : ~ ÷ X ) induced Since But then of the it f o l l o w s induces ~l(X× is s u r j e c t i v e ~ Im(6,) this (e,l) X* is t r £ v i a l .

Is t h e be t h e S = ~(X , which P(X) is t h e tangent in g e n e r a l Let tions. is t h e The are when to { (p,£) and let secant closure , and morphism the line denote from through x of ) of of by ~< 2n - 1 tangent is the this Then parametrized X , along , consisting projectivized lines; X × X divisor dimension belong respondence, ~< 2n limits has F = of T = ~(P(X) dimension T c G all blow-up exceptional , and These n to a m o r p h i s m : X x X + G where to (x,y) varieties. dimension The pr a pair secant bundle singular, diagonal blowing-up; S c G is an all secant the closed When X of not X [38]).

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 .

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