By David Mumford

From the experiences: "Although a number of textbooks on smooth algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the pink publication of sorts and schemes, now as sooner than some of the most very good and profound primers of contemporary algebraic geometry. either books are only real classics!" Zentralblatt

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**Extra resources for Algebraic Geometry I: Complex Projective Varieties**

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9, D e f . 3 below). W e c o n s i d e r t h e ~(Mo) c A ( R ) , with that space V~ ~ is a b r a n c h i n g of X(R). := { 6 n l n 6 ~ }. T h e n M Proof. V c V ° the triangulation We conclude Assume tains a strictly ing to z e r o point ~ 6X. 10. d. §5 - P r o p e r In t h i s and maps section a "map" else means following exhaustions of M w i t h a weakly a weakly semialgebraic semialgebraic to the I. 16, given map between It s u f f i c e s N B. subset space spaces, every iv) pull subspace back f' by an a r b i t r a r y (cf.

T h e of M w i t h exhaustion. 17) describes a 6 I we define set ~(M) a running to t h e : M ~ I fully = a}. a £ I is c a l l e d a 6~(M). ~ < a}. := {x 6 M 1 q ( x ) index function a 6 I, = {x 6 M I ~ ( x ) Definition An the index of the set M i n t o n o n e m p t y disjoint subsets. 19. algebraic finite of M w i l l be v e r y useful L e t X be a s i m p l i c i a l space (Ex. 7). Y which We in t h e complex, consider are closed following. regarded as a w e a k l y the exhaustion of X b y t h e in X, every Y being indexed semi- by 14 itself.

4. map in R to M. p a t h y in M c a n be ~ M.