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

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

From the reports of the 1st printing, released as quantity 23 of the Encyclopaedia of Mathematical Sciences:
"This volume... involves papers. the 1st, written through V.V.Shokurov, is dedicated to the idea of Riemann surfaces and algebraic curves. it really is a great evaluation of the speculation of family among Riemann surfaces and their versions - complicated algebraic curves in advanced projective areas. ... the second one paper, written via V.I.Danilov, discusses algebraic types and schemes. ...
i will suggest the ebook as a good advent to the fundamental algebraic geometry."
European Mathematical Society publication, 1996

"... To sum up, this ebook is helping to profit algebraic geometry very quickly, its concrete kind is pleasant for college kids and divulges the great 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

If all the A-periods of a holomorphic form w on S are zero, then w = O. Riemann's bilinear relations play an important role in the theory of abelian varieties (see Example 2 in Chap. 3, Sect. 3). 8. Meromorphic Differentials; Canonical Divisors Definitions. A meromorphic differential on a Riemann surface S is a holomorphic differential w on some open subset U c S whose complement S - U is discrete in S, with the property that locally w = fdz, where f is a meromorphic function with poles in S - U.

Any differential I-form w on 8 can be written locally as wlu = fdz + gdz = (f + g)dx + (f - g)H dy, where f and 9 are complex-valued functions on an open set U c 8, and z = x + A y is a local coordinate on U. A differential w is said to be differentiable if, for every local representation w = fdz + gdz, the functions f and 9 are differentiable. We denote by Al the (infinite-dimensional) complex vector space of differentiable I-forms on 8. A V. V. Shokurov 46 form W is of type (1,0) (respectively, (0,1)) if, locally, W = idz (respectively, W = idz).

L :-;uriaces and Algebraic Curves 27 Remark 2. If f == c is a constant function then ordp f = 0 for c i- 0; it is convenient to consider that ordp 0 = +00. 5. Topological Properties of Mappings of Riemann Surfaces. All propo- sitions in this subsection have a local nature. Hence they are easy to derive from the corresponding facts belonging to the theory of analytic functions of one variable. (For instance, the first two propositions follow in an obvious way from the Lemma on normal form stated in the preceding subsection).

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