By Daniel Perrin (auth.)

ISBN-10: 2759800482

ISBN-13: 9782759800483

Aimed basically at graduate scholars and starting researchers, this publication offers an advent to algebraic geometry that's relatively compatible for people with no past touch with the topic and assumes simply the traditional historical past 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 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 basic instruments of recent algebraic geometry: size; singularities; sheaves; types; and cohomology. The remedy makes use of as little commutative algebra as attainable through quoting with no evidence (or proving in basic terms in specific circumstances) theorems whose evidence isn't really precious in perform, the concern being to improve an figuring out of the phenomena instead of a mastery of the strategy. a variety of workouts is equipped for every subject mentioned, and a variety of difficulties and examination papers are gathered in an appendix to supply fabric for additional study.

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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.

It is also a bijection whose inverse is given by (x1 , . . , xn ) → (1, x1 , . . , xn ). Moreover, since the hyperplane H is a projective space of dimension n − 1, the foregoing gives a description of projective space Pn (k) of dimension n as being a disjoint union of an aﬃne space k n of dimension n and a projective space H of dimension n − 1. Alternatively, we have embedded a copy of aﬃne space k n in a projective space of the same dimension. The points of k n are said to be “at ﬁnite distance” and the points of H are said to be “at inﬁnity”.

Xn ]. In small dimensions we will mostly use variables x, y, z, t and take the hyperplane t = 0 to be the hyperplane at inﬁnity. The ﬁrst diﬀerence with aﬃne sets is that the polynomials F in the ring k[X0 , . . , Xn ] no longer deﬁne functions on projective space since their value at a point x depends on the chosen system of homogeneous coordinates. For example, if F is homogeneous of degree d, then F (λx0 , λx1 , . . , λxn ) = λd F (x0 , x1 , . . , xn ). However, we can deﬁne zeros of polynomials in the following way.

### Algebraic Geometry: An Introduction by Daniel Perrin (auth.)

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