By Oscar Zariski

ISBN-10: 0387053352

ISBN-13: 9780387053356

The most objective of this e-book is to offer a totally algebraic method of the Enriques¿ category of soft projective surfaces outlined over an algebraically closed box of arbitrary attribute. This algebraic procedure is without doubt one of the novelties of this ebook one of the different glossy textbooks dedicated to this topic. chapters on floor singularities also are incorporated. The ebook will be helpful as a textbook for a graduate direction on surfaces, for researchers or graduate scholars in algebraic geometry, in addition to these mathematicians operating in algebraic geometry or similar fields"

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**Extra resources for Algebraic Surfaces**

**Example text**

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 Surfaces by Oscar Zariski

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