By Daniel Perrin

ISBN-10: 1848000561

ISBN-13: 9781848000568

Aimed essentially at graduate scholars and starting researchers, this ebook offers an creation to algebraic geometry that's really appropriate for people with no earlier touch with the topic and assumes basically the normal heritage of undergraduate algebra. it's built from a masters direction given on the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field.

The e-book starts off with easily-formulated issues of non-trivial ideas – for instance, Bézout’s theorem and the matter of rational curves – and makes use of those difficulties to introduce the elemental instruments of contemporary algebraic geometry: measurement; singularities; sheaves; kinds; and cohomology. The remedy makes use of as little commutative algebra as attainable by way of quoting with out evidence (or proving merely in particular circumstances) theorems whose evidence isn't worthy in perform, the concern being to improve an knowing of the phenomena instead of a mastery of the strategy. a variety of workouts is supplied for every subject mentioned, and a range of difficulties and examination papers are gathered in an appendix to supply fabric for additional research.

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**Extra info for Algebraic Geometry: An Introduction (Universitext)**

**Example text**

1. Let V be a subset of Pn . We deﬁne the ideal of V by the formula Ip (V ) = {F ∈ k[X0 , . . 1)}. 2. 1 (cf. 2) b) The operation Ip is decreasing. c) If V is a projective algebraic set, then Vp (Ip (V )) = V . If I is an ideal, then I ⊂ Ip (Vp (I)). d) We have Ip (Pn ) = (0) and I(∅) = k[X0 , . . , Xn ]. 3. Irreducibility. The deﬁnitions and results of Chapter I can be easily translated mutatis mutandis into projective geometry. Assume now that the ﬁeld k is algebraically closed. There is then a projective version of the aﬃne Nullstellensatz.

We denote the set of regular maps from V to W by Reg(V, W ). 2. It is clear that we obtain in this way a category: the identity is a morphism, as is the composition of two morphisms. All the usual notions— isomorphisms, automorphisms, and so forth—therefore apply. We note that morphisms are continuous maps for the Zariski topology (which is to say that the preimage of an algebraic set under a morphism is again an algebraic set), but the converse is false (for example, any bijective map from k to k is continuous for the Zariski topology but is not necessarily polynomial).

Xn ] and x ∈ Pn . We say that x is a zero of F if F (x) = 0 for any system of homogeneous coordinates x for x. We then write either F (x) = 0 or F (x) = 0. If F is homogeneous, it is enough to check that F (x) = 0 for any system of homogeneous coordinates. If F = F0 + F1 + · · · + Fr , where Fi is homogeneous of degree i, then it is necessary and suﬃcient that Fi (x) = 0 for all i. 30 II Projective algebraic sets Proof. Only the last statement needs to be proved. If F (λx) = λr Fr (x) + · · · + λF1 (x) + F0 (x) = 0 for any λ, then since k is inﬁnite all the values Fi (x) vanish.

### Algebraic Geometry: An Introduction (Universitext) by Daniel Perrin

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