By David Mumford
From the reports: "Although a number of textbooks on sleek algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the crimson ebook of sorts and schemes, now as prior to essentially the most very good and profound primers of contemporary algebraic geometry. either books are only actual classics!" Zentralblatt
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Additional info for Algebraic Geometry I: Complex Projective Varieties
If the vector field is projected onto its real part we get 41 HANDLEBODY STRUCTURES a real normal vector field to £ except at °°. On the subset ? of «-, this complex vector field is equal to vector field on the set |t*| = 1 f (t ) (8/3£ ). We modify the t , N 0/9£ f ). |t'| < 1 by defining it to be taking the real part of this vector field and using it to push 2 £ = -N. Now, £ off of itself we find that The fiber type of S over a point a € C1P depends on the multipli- city of a m, n, d denote these multiplicities (if a polynomial is as a root of each of the polynomials: g 9 (t), g~(t), A(t).
G 9 (t), X(t) . g (t), g«(t) would make the equation unsuitable for Secondly, and what is more important for our purposes, we wish to use the equations to compare the topology of different elliptic surfaces. For this purpose, the compactification must either be nonsingular or at worst have singularities which are rational double points (see appendix to §0 for the definition). Actually, both problems can be solved by the follow- ing procedure: given an equation of the form: y z = x where g&(t), g£(t) constant.
It is well known that the connected component of the identity, 2 2 Diffn(T ) , has T , acting on itself by translations, as a deformation 2 2 retract, and that Diff(T )/Diffn(T ) is isomorphic to SL(2,Z). Thus up to -1 smooth isotopy we may assume that S^So is of the form 0 x £ •> 6 x A? + b(0) where 2 T:D X T 2 -> D 2 x T 2 6 € S 1 , A € SL(2,Z) ,£, b(9) 6 T 2. Define x:z x 5 ->> z x A" 1 ^, Z € D2, £ € T 2 . Thus by £ £~ T:6 x £ -> 0 x £ + b(9). , £ 1 £ ? x:S X T -> S X T 2 1 is the identity outside of jugating with £ I x T we get that where I f is any interval in 1 £ 9 x£,:8S -> 3S S .