By Daniel Perrin (auth.)
Aimed basically at graduate scholars and starting researchers, this booklet presents an creation to algebraic geometry that's relatively appropriate for people with no prior touch with the topic and assumes purely the normal heritage of undergraduate algebra. it truly is constructed from a masters path given on the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field.
The booklet begins 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 basic instruments of contemporary algebraic geometry: size; singularities; sheaves; kinds; and cohomology. The remedy makes use of as little commutative algebra as attainable by way of quoting with no facts (or proving merely in distinctive instances) theorems whose evidence isn't really helpful in perform, the concern being to strengthen an realizing of the phenomena instead of a mastery of the strategy. quite a number workouts is supplied for every subject mentioned, and a range of difficulties and examination papers are gathered in an appendix to supply fabric for extra study.
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Extra info for Algebraic Geometry: An Introduction
2. Let X be a topological space equipped with a basis of open sets U, let F be a sheaf and let G be a presheaf on X. We assume F(U ) = G(U ) for every U ∈ U. Then F = G + (cf. c). In the case in hand we therefore seek to deﬁne Γ (D(f ), OV ). Since D(f ) is the set of points where the function f does not vanish, it is natural to include the inverse function f −1 along with the polynomial functions on V in the set of sections Γ (D(f ), OV ). More precisely, we consider the restriction homomorphism r : Γ (V ) → F(D(f ), k), where F(D(f ), k) denotes the ring of all functions from D(f ) to k.
We can of course deﬁne sheaves of other structures (groups, modules or k-algebras) in a similar way. 5. A ringed space is a topological space X equipped with a sheaf of rings. This sheaf is called the structural sheaf of X and is traditionally denoted by OX . “Morally” this sheaf is the sheaf of “good” functions on X and we have therefore assumed that a sum or product of good functions is still a good function. 7. From now on we ﬁx an algebraically closed ﬁeld k. Unless otherwise speciﬁed, the structural sheaf of all ringed spaces considered will be a sheaf of k-valued functions and we will assume it is a sheaf of k-algebras containing the constant functions.
A quadric in P3 (k) is a projective algebraic set of the form Q = V (F ), where F is an irreducible polynomial of degree 2 in X, Y, Z, T and hence gives rise to a quadratic form on k4 which we assume to be non-degenerate. a) Prove that if Q = V (F ) is a quadric, then there is a homography h such that h(Q) = V (XT − Y Z). We assume from now on that Q is of this form. b) Prove that Q contains two families of lines both of which are indexed by P1 . Prove that a unique line from each family passes through any point of Q, that two lines in the same family are disjoint and that two lines in diﬀerent families meet at a unique point.