By David Mumford

ISBN-10: 0387076034

ISBN-13: 9780387076034

From the experiences: "Although a number of textbooks on sleek algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the crimson booklet of sorts and schemes, now as earlier than some of the most first-class and profound primers of recent algebraic geometry. either books are only precise classics!" Zentralblatt

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**Extra info for Algebraic Geometry I: Complex Projective Varieties**

**Sample text**

2 It is an easy argument in bilinear algebra to see that there exists a basis of k n+1 such that the matrix of β with respect to this basis is a diagonal matrix with 1 and 0 on its diagonal. By permuting the basis we may assume that the ﬁrst entries of the diagonal are 1’s. e. the number of 1’s. In particular, r is independent of our choice of the basis. 2 is irreducible if and only if r > 2. 68. The polynomial X02 + · · · + Xr−1 2 2 n closed subspace V+ (X0 + · · · + Xr−1 ) of P (k) is irreducible if and only if r = 2.

Xn ). 2) ∼ Φi : F → K(T0 , . . , Ti , . . , Tn ), Φi (f ) f → . g Φi (g) Xn 0 Often, we will identify K(T0 , . . , Ti , . . , Tn ) with the subring K( X Xi , . . , Xi ) of the ﬁeld K(X0 , . . , Xn ). 2) can also be described as follows. Let fg ∈ F with f, g ∈ K[X0 , . . , Xn ]d for some d. Set f˜ = Xfd and g˜ = Xgd . i i f f˜ Xn 0 Then f˜, g˜ ∈ K[ X Xi , . . , Xi ] and Φi ( g ) = g ˜. 20) Deﬁnition of the projective space Pn (k). The projective space Pn (k) is an extremely important prevariety within algebraic geometry.

Xn ). We call the xi the homogeneous coordinates on Pn (k). 28 1 Prevarieties To endow Pn (k) with the structure of a prevariety we will assume from now on that k is algebraically closed. The following observation is essential: For 0 ≤ i ≤ n we set Ui := { (x0 : . . : xn ) ∈ Pn (k) ; xi = 0 } ⊂ Pn (k). This subset is well-deﬁned and the union of the Ui for 0 ≤ i ≤ n is all of Pn (k). , ,... xi xi xi (x0 : . . : xn ) → . Via this bijection we will endow Ui with the structure of a space with function, isomorphic to (An (k), OAn (k) ), which we denote by (Ui , OUi ).

### Algebraic Geometry I: Complex Projective Varieties by David Mumford

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