By Daniel Perrin (auth.)

Aimed essentially at graduate scholars and starting researchers, this publication presents an creation to algebraic geometry that's fairly appropriate for people with no prior touch with the topic and assumes merely the traditional heritage of undergraduate algebra. it's built from a masters path given on the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field.

The ebook begins with easily-formulated issues of non-trivial strategies – 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 therapy makes use of as little commutative algebra as attainable through quoting with out facts (or proving merely in unique circumstances) theorems whose evidence isn't priceless in perform, the concern being to boost an knowing of the phenomena instead of a mastery of the process. a number routines is equipped for every subject mentioned, and a variety of difficulties and examination papers are gathered in an appendix to supply fabric for extra study.

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**Example text**

Then (cf. 2) there exists a non-zero f ∈ Γ (W ) which vanishes on X. But then f ϕ = ϕ∗ (f ) = 0, which is a contradiction. 2. 12. We note that the conditions are dual to each other because of the contravariance of Γ . Be careful, however: ϕ∗ injective does not imply ϕ surjective (consider the projection of the hyperbole XY = 1 on the x-axis). We ﬁnish this section by showing that when the ﬁeld k is algebraically closed, the situation is as good as it could be. 13. Assume that k is algebraically closed.

Desargues, but was mostly developed in the xixth century (by Monge, Poncelet, Klein and others). We have known that it is the natural setting for most geometries since Klein’s Erlangen programme (1872). In algebraic geometry, it is also the setting which gives the most satisfying results. However, aﬃne space remains important since it is a local model of projective space. 1 Projective space a. Deﬁnition Let n be an integer 0 and let E be a k-vector space of dimension n + 1. We introduce the equivalence relation R on E − {0}: xRy ⇐⇒ ∃ λ ∈ k ∗ , y = λx.

Show that C is an aﬃne algebraic set and calculate I(C). Show that Γ (C) is isomorphic to the ring of polynomials k[T ]. 8) Assume that k is algebraically closed. Determine the ideals I(V ) of the following algebraic sets. V (XY 3 + X 3 Y − X 2 + Y ), V (X 2 Y, (X − 1)(Y + 1)2 ), V (Z − XY, Y 2 + XZ − X 2 ). II Projective algebraic sets Throughout this chapter, k will be a commutative ﬁeld. 0 Motivation We have already seen the main reason for introducing projective space in the Introduction when discussing B´ezout’s theorem.