By Joe Harris

This publication is predicated on one-semester classes given at Harvard in 1984, at Brown in 1985, and at Harvard in 1988. it's meant to be, because the identify indicates, a primary creation to the topic. on the other hand, a couple of phrases are so as concerning the reasons of the e-book. Algebraic geometry has built significantly over the past century. in the course of the nineteenth century, the topic was once practiced on a comparatively concrete, down-to-earth point; the most items of research have been projective kinds, and the concepts for the main half have been grounded in geometric structures. This strategy flourished throughout the center of the century and reached its fruits within the paintings of the Italian institution round the finish of the nineteenth and the start of the 20 th centuries. finally, the topic was once driven past the boundaries of its foundations: by way of the top of its interval the Italian institution had advanced to the purpose the place the language and methods of the topic may possibly now not serve to precise or perform the guidelines of its most sensible practitioners.

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**Example text**

First of all, if L c K is a subfield, we will write A"(L) for the subset c Kn =k. /Z; E L whenever defined—that is, points that may be written as EZ0 , Zn] with Zi e L. All of what follows applies to projective varieties, but we will say it only in the context of affine ones. We say that a subvariety X c Pq is defined over L if it is the zero locus of polynomials L(z i , , zn) E L [z 1 , , zn]. For such a variety X, the set of points of X defined over L is just the intersection X n An(L). We should not, however, confuse the set of points of X defined over L with X itself; for example, the variety 0 is defined over IR and has no points in A 2c defined by the equation x 2 + y 2 + I defined over 11 but it is not the empty variety.

PROOF. Let F(X) be the homogeneous polynomial defining the hypersurface Y; say the degree of F is d. 14 to the regular functions G/F, where G ranges over homogeneous polynomials of degree d on Pit, to deduce that X is a point. 13 raises the question of what the image of an affine or quasi-projective variety X may be under a regular map f: X -- P". The first thing to notice is that it does not have to be a quasi-projective variety. The primary example of this is the map f: A' --* A 2 given by f(x, y) = (x, xy).

Thus, for X LI1 -- lp e X: f(p) 0 , where f ranges over polynomials; these are called the distinguished open subsets of X. Similarly, for X c ll=" projective, a basis is given by the sets UF --- f p e X: F(p) 0} for F a homogeneous polynomial; again, these open subsets are called distinguished. This is the topology we will use on all the varieties with which we deal, so that if we refer to an open subset of a variety X without further specification, we will mean the complement of a subvariety. Implicit in our use of this topology is a fundamentally important fact: inasmuch as virtually all the constructions of algebraic geometry may be defined algebraically and make sense for varieties over any field, the ordinary topology on P', (or, as it's called, the classical or analytic topology) is not logically relevant.