Algebraic Geometry: An Introduction (Universitext)
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Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject; it assumes only the standard background of undergraduate algebra. The book starts with easily-formulated problems with non-trivial solutions and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.
correspond to polynomials of degree 1. 3) Consider V ⊂ k n . The projection f from V to k p , p n, given by ϕ(x1 , . . . , xn ) = (xi1 , . . . , xip ), is a morphism. 4) Let V be the parabola V (Y −X 2 ) and let f be the projection ϕ : V → k given by ϕ(x, y) = x. Then f is an isomorphism, whose inverse is given by x → (x, x2 ). 6 An introduction to morphisms 21 5) The map ϕ : k → V (X 3 + Y 2 − X 2 ), given by the parameterisation x = t2 − 1, y = t(t2 − 1) (obtained by intersecting with the
constructions, such as homomorphisms, kernels, images, exact sequences, etc., are also possible with OX -modules. Deﬁnition 6.4. Let F, G be two OX -modules. A homomorphism f : F → G is given by the data of OX (U )-linear maps for every U , f (U ) : F(U ) → G(U ) which are compatible in the obvious way with restrictions. We can then deﬁne the kernel sheaf of f by the formula (Ker f )(U ) = Ker(f (U )), and we say that f is injective if f (U ) is injective for all U , or, alternatively, if Ker f =
point x is of rank n − dim V , then the point x is smooth in V . 3 Regular local rings We start by recalling an algebraic result. Proposition-Deﬁnition 3.1. Let A be a local Noetherian ring, let m be its maximal ideal and set k = A/m. The quotient m/m2 m ⊗A k is a k-vector space and dimk m/m2 dimK A (the Krull dimension of A, cf. Chapter IV, 1.5). We say that A is regular if dimk m/m2 = dimK A. 94 V Tangent spaces and singular points Proof. The fact that m/m2 is a k-vector space is
the points are of multiplicity 1. Proposition 1.3. Let (Z, OZ ) be a ﬁnite scheme. For any subset V in Z, Γ (V, OZ ) = P ∈V OZ,P . Conversely, if we assign to every point of a ﬁnite set Z a local ﬁnite k-algebra, then the above formula deﬁnes a ﬁnite scheme structure on Z. Proof. This is clear: we associate to any section over V its restriction to each of the (open) points P ∈ V , and the gluing condition is empty in this case. Deﬁnition 1.4. If Z is a ﬁnite scheme and A = Γ (Z, OZ ), then we
: P1 → C given by the formula ϕ(λ, µ) = (4µ2 (λ+µ)2 , 4µ2 (λ−µ)2 , (λ2 −µ2 )2 ). Prove that this morphism is an isomorphism except at the points (1, 0), (−1, 1) and (1, 1) whose images are the points P, Q and R and that the inverse of ϕ outside of P, Q, R is given by the map ϕ−1 (x, y, t) = ((x − y)t, xy). e) Construct the real part of C in the aﬃne plane T = 0. VII Sheaf cohomology 0 Introduction We return for a moment to the proof of B´ezout’s theorem. Given Z = V (F, G) we had to calculate