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 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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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. 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 .

A morphism differentials the to P r o j e c t i v e singularities (A). pm of M a p p i n g s : X + ~m f*Z f*~ was theorem (but p o s s i b l y However gularities shows is the n o r m a l i - conjecture. In this tive f result connectedness is a s u b v a r i e t y and Hartshorne §5. , A related Corollary > codim(Y,Z) Hironaka n in w h i c h [58]. 1. embeds X as sheaf of AX ideal I/I 2 It is an o p e n Given a closed in follows (as w e l l as a morphism f subscheme X Xy X , that f a closed) - X + Y is AX of 1 g2X/Y then unramified subscheme , the of diagonal X ×y X can be if a n d X ×y X map If I identified only (cf if Ax [27, IV.

Ample proved by suggested between normal empty) observed 199]) give the complete bundle, provided that that examples of counter-examples is a c l o s e d that to that 1 ÷ ~X f Let f 1 ~X/Y is s u r j e c t i v e . e. topology. The X theorem philosophy must to s t u d y here sin- is t h a t occur. be a c o m p l e t e be an u n r a m i f i e d : X + Y variety morphism. f is u n r a m i f i e d is zero. is u n r a m i f i e d that connectedness Space. of d i m e n s i o n If 2n > m , embedding. a morphism differentials the to P r o j e c t i v e singularities (A).

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