By David Mumford

ISBN-10: 0387076034

ISBN-13: 9780387076034

From the reports: "Although a number of textbooks on glossy algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the crimson ebook of types and schemes, now as sooner than some of the most very good and profound primers of contemporary algebraic geometry. either books are only real classics!" Zentralblatt

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**Sample 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 I: Complex Projective Varieties by David Mumford

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