Download Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds by I. R. Shafarevich (auth.), I. R. Shafarevich (eds.) PDF

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

From the studies of the 1st printing, released as quantity 23 of the Encyclopaedia of Mathematical Sciences:
"This volume... involves papers. the 1st, written by means of V.V.Shokurov, is dedicated to the idea of Riemann surfaces and algebraic curves. it truly is a great assessment of the idea of kinfolk among Riemann surfaces and their types - advanced algebraic curves in advanced projective areas. ... the second one paper, written by means of V.I.Danilov, discusses algebraic types and schemes. ...
i will suggest the ebook as an exceptional advent to the fundamental algebraic geometry."
European Mathematical Society e-newsletter, 1996

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

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

Example text

Example 2. Let f: 8 1 - t 8 2 be a nonconstant mapping of Riemann surfaces. -l(p) whose support is the fibre f-1(p). By additivity, this defines a homomorphism f*: Div 8 2 -t Div 8 1 , L aiPi L ad*(pi)' f-+ The divisor R ~ I: rp(f) . P E Div 8 1 , where rp(f) is the ramification index of f at p, is called the ramification divisor of f. Example 3. Now let f be a nonconstant meromorphic function on a Riemann surface 8. (p)=oo are called the divisor of zeros and the divisor of poles of f, respectively.

The fundamental group is also useful for the description of finite mappings. If f: SI ----* S2 is a finite mapping of Riemann surfaces then we have a finite unramified covering f: SI - f- l (11) ----* S2 - 11, where the branch locus 11 C S2 is the discrete subset above which the ramification points lie. Conversely: Proposition. Let 11 C S2 be a discrete subset. A finite unramified covering U ----* 8 2 - 11 has a unique continuation to a (possibly ramified) finite mapping SI ----* S2, where SI =:J U.

Riemann Surfaces and Algebraic Curves of ~f of dz oz and 8f 47 ~f 0/ = o~dz. OZ These mappings are «~:-linear and have natural continuations to C-linear maps d, 0, 8: Al ----+ A2. Locally, d(fdz + gdz) ~f df i\ dz + dg i\ dz = (~~ o(fdz + gdz) and - o(fdz + gdz) - ~~) dz i\ dz, ~f of i\ dz + og i\ dz = ~~ dz i\ dz, def - = of i\ dz - + og i\ dz = - of ozdz i\ dz. The maps d, 0, 8: Ai ----+ Ai+I, for i :2: 2, are defined in a similar way. However, for Riemann surfaces they are trivial, since Ai = 0 for i :2: 3.

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