By James R. Milgram
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Extra resources for Algebraic and Geometric Topology
It follows that ' W X0 ! W0 is a morphism. In the same way one proves that W W0 ! X0 is a morphism. The preceding proposition has an important consequence. 6. Given a projective (or affine) variety X , the following two conditions are equivalent. t1 ; : : : ; td / for some integer d . (2) There is a dense open subset X0 subset U0 Ad . 6 is said to be rational. In particular condition (2) is the precise statement of the fact that a rational variety X can be parameterized by d independent variables (cf.
Y is surjective. w/ D ;, by what we have just seen one would have the contradiction mw D KŒX . • (Finiteness is a local property) A morphism ' W X ! U˛ / ! U˛ is a finite morphism for each index ˛. 11 (Birational equivalence of a projective variety with a hypersurface). Among the transformations between two projective spaces one has in particular the projections. Projecting the points of P n from a subspace Sk of P n onto a subspace Sk 0 skew to it and of dual dimension (that is, k C k 0 D n 1) one obtains a rational mapping ' W P n !
X /. Thus one has an isomorphism KŒX Š R; which expresses the coordinate ring of X as a ring of polynomial functions (defined on all of X) with values in K. Now let An and Am be two affine spaces with coordinate rings KŒY1 ; : : : ; Yn and KŒT1 ; : : : ; Tm respectively. We say that a map W An ! y1 ; : : : ; yn / for j D 1; : : : ; m. Y1 ; : : : ; Yn /; j D 1; : : : ; m; 22 Chapter 2. Algebraic Sets, Morphisms, and Rational Maps are its equations. If X An , W Am are algebraic sets, we say that a map W X !
Algebraic and Geometric Topology by James R. Milgram