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
H ° ( lDm , 0 1 ~ ( i ) ) that at o n l y intersections incidence 0p(1) imply X c pm in N conjecture Recall X the second by theses Hartshorne's X is T h e o r e m of Corollary , where defined this for c o m p l e t e in t h e p r o o f assertion to points. is a h y p e r s u r f a c e , To verify if c a n be t a n g e n t linearly intersection, the 3n > 2 ( m - i), his conjecture. theorem space closed X for e x a m p l e , Motivated of by the sufficiently  w a s small led on the conjecture: subvariety is l i n e a r l y on tangencies Equivalently, shows, Hartshorne following then one normal.
Normality. variety X c pm of a is l i n e a r l y (non-degenerate) X c pm is l i n e a r l y normal embedding normal of if a n d map are that a subvariety codimension basis From z is an a m p l e the p r o j e c t i o n Alternatively, the point (cf. [13, Exp. 4. H ° ( lDm , 0 1 ~ ( i ) ) that at o n l y intersections incidence 0p(1) imply X c pm in N conjecture Recall X the second by theses Hartshorne's X is T h e o r e m of Corollary , where defined this for c o m p l e t e in t h e p r o o f assertion to points.
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. (X- Z) ×X ~ (B), we will frequently need to know that t e r m i n o l o g y of algebraic geometry): and being connected, normal v a r i e t y has this property. topology for then §i. Statement an integer X Let any space is irreducible. Generic Linear Sections X be a variety, X and is irreducible, [ii].