By Daniel Perrin (auth.)

Aimed essentially at graduate scholars and starting researchers, this booklet presents an creation to algebraic geometry that's relatively appropriate for people with no earlier touch with the topic and assumes simply the normal historical past of undergraduate algebra. it truly is constructed from a masters direction given on the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field.

The booklet starts off with easily-formulated issues of non-trivial recommendations – for instance, Bézout’s theorem and the matter of rational curves – and makes use of those difficulties to introduce the elemental instruments of recent algebraic geometry: measurement; singularities; sheaves; forms; and cohomology. The remedy makes use of as little commutative algebra as attainable by way of quoting with no evidence (or proving in simple terms in certain situations) theorems whose evidence isn't useful in perform, the concern being to improve an figuring out of the phenomena instead of a mastery of the method. a number of routines is supplied for every subject mentioned, and a range of difficulties and examination papers are accumulated in an appendix to supply fabric for additional study.

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**Sample text**

Xn ). We note that the elements xi are not all 0 and if λ ∈ k is = 0, then (λx0 , λx1 , . . , λxn ) is another system of homogeneous coordinates for x, which justiﬁes our terminology. 2. 1) When k = R or C, projective space has a natural topology, namely the quotient of the topology on k n+1 −{0}. Projective space is then easily checked to be compact and connected. 2) The fact that the projective space associated to k n+1 is of dimension n corresponds to the fact that lines passing through the origin are contracted to points.

C) Points are projective algebraic sets: consider x = (x0 , x1 , . . , xn ) ∈ Pn . One of the component xi —for example, x0 —is not 0, so we can assume x0 = 1. We then have {x} = Vp (X1 − x1 X0 , . . , Xn − xn X0 ). d) If n = 2, projective plane curves are deﬁned by homogeneous equations: Y 2 T − X 3 = 0, X 2 + Y 2 − T 2 = 0, . . 5. As in the aﬃne case, the following hold. a) The map Vp is decreasing. b) An arbitrary intersection or ﬁnite union of projective algebraic sets is a projective algebraic set, so there is a (Zariski) topology on Pn whose closed sets are the projective algebraic sets.

1 there is a non-zero polynomial d ∈ k[X] and polynomials A, B ∈ k[X, Y ] such that d = AF +BG. 3). We leave the details of the proof to the reader: we simply apply B´ezout’s (elementary) theorem to the principal ring k(X)[Y ] and clear denominators. 1. 3 d(x) = 0 and hence there are a ﬁnite number of possible values x. The same reasoning applied to y shows that the intersection is ﬁnite. 3 we see that a ﬁnite number of these monomials generate k[X, Y ]/(F, G). 4. 3 is the resultant of F and G, considered as polynomials in Y .