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

ISBN-10: 3540108335

ISBN-13: 9783540108337

ISBN-10: 354038720X

ISBN-13: 9783540387206

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

Example text

4. 3 hold (cf. 1 and the assertion of [19]). This may be proved as before by pas- A ~ (pm)r to a linear space A d i f f e r e n t proof of Theorem Lm _ c pr(m+l)-i NOTES. 1(A) was given by Barth [19]. 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.

Theorem then every of the b r a n c h space, R1 covering ~ £ connected. with normal, identifying induces which to Hence X ~ S* Zl(X) loci to the c o r r e s p o n d i n g variety S | of T h e o r e m of p r o j e c t i v e take = d} (l,e) is n o n - e m p t y , on the p u r i t y ×~n if then is a b r a n c h e d proof ; one may at = ~I(S*) × ~l(X) ramification if (2) X) ~l(S*) and an a l t e r n a t i v e S* possible non-singular, to g i v e of d i m e n s i o n homomorphism element, of the f(S) S , one h a s a non-trivial S c X Rd_ 1 = {x • X l e f ( x ) ÷ ~l(S*x image set over the natural this in t h e if of of_ ~I(S*) is s u r j e c t i v e .

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. all k = n incidence { (x,L)ITx realizes 1 second X c ~m tangent to X dimension X* according ~ m k = m - 1 conrespec- × pm* pm-n-i variety But and immediate correspondence ~ X a are -bundle c pm* to - n - is the 1 over X the image theorem, , and the , and the result | remark that achieved for Proof Theorem of first , consider projection dimP under .

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Algebraic Geometry: Proceedings of the Midwest Algebraic Geometry Conference, University of Illinois at Chicago Circle, May 2 – 3, 1980 by Igor Dolgachev, Anatoly Libgober (auth.), Anatoly Libgober, Philip Wagreich (eds.)


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