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Read e-book online Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds PDF

By I. R. Shafarevich (auth.), I. R. Shafarevich (eds.)

ISBN-10: 3540637052

ISBN-13: 9783540637059

ISBN-10: 3642578780

ISBN-13: 9783642578786

From the studies of the 1st printing, released as quantity 23 of the Encyclopaedia of Mathematical Sciences:
"This volume... contains papers. the 1st, written by means of V.V.Shokurov, is dedicated to the speculation of Riemann surfaces and algebraic curves. it truly is a good review of the idea of kin among Riemann surfaces and their versions - complicated algebraic curves in advanced projective areas. ... the second one paper, written via V.I.Danilov, discusses algebraic forms and schemes. ...
i will suggest the publication as an exceptional advent to the fundamental algebraic geometry."
European Mathematical Society publication, 1996

"... To sum up, this e-book is helping to profit algebraic geometry very quickly, its concrete type is agreeable for college students and divulges the wonderful thing about mathematics."
Acta Scientiarum Mathematicarum, 1994

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Extra resources for Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes

Sample text

In this way we also show the existence and uniqueness of the expansion of a proper fraction into partial fractions in the complex sense. Thus M(ClP'l) ':::::' C(z), where C(z) is the field of rational functions of one variable z. Hence meromorphic functions constitute a natural generalization of rational functions. Furthermore, these notions coincide for any compact Riemann surface, once the rationality of a function is suitably defined (see Sect. 5). Example 2. Let J: 3 1 -+ 3 2 be a nonconstant mapping of Riemann surfaces.

Then g(8) = ~n -1. Hence on any orient able compact surface there is a Riemann surface structure, a hyperelliptic one, for instance. Example 2. Let 8 be a compact Riemann surface of genus g, and 8 ---+

W2(t2). This form is called the tensor product of WI and W2' The tensor product of any number of differentials is defined in a similar way. Locally, every differential W can be written as a linear combination of tensor products of the differentials dz and dE, with functions as coefficients. If each product has i factors then W is said to be an i-form. Further, if all the coefficients are differentiable functions then we say that W is differentiable. As with multilinear functions, one can impose conditions of symmetry, skew- or Hermitian-symmetry, and the like.

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Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes by I. R. Shafarevich (auth.), I. R. Shafarevich (eds.)


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