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... includes papers. the 1st, written via V.V.Shokurov, is dedicated to the speculation of Riemann surfaces and algebraic curves. it really is a very good evaluation of the speculation of kinfolk among Riemann surfaces and their versions - advanced algebraic curves in complicated projective areas. ... the second one paper, written via V.I.Danilov, discusses algebraic kinds and schemes. ...

i will be able to suggest the ebook as a good advent to the fundamental algebraic geometry."

European Mathematical Society e-newsletter, 1996

"... To sum up, this ebook is helping to benefit algebraic geometry very quickly, its concrete sort is pleasant for college kids and divulges the wonderful thing about mathematics."

Acta Scientiarum Mathematicarum, 1994

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**Example text**

2. Triangulability. A triangle on a Riemann surface S is a homeomorphic image T of an ordinary Euclidean triangle with the usual topology. The image of a vertex is called a vertex of T, and the image of a side is called an edge. A triangulation of S is a family {Ti} of triangles on S such that 1. Riemann Surfaces and Algebraic Curves 37 (a) S = UTi; (b) if two triangles meet then their intersection consists either of a common vertex or of a common edge; (c) if {Ti} is not a finite family, then we demand that it should be locally finite; this amounts to saying that only finitely many triangles have a common vertex and that their union defines a neighbourhood of that vertex (cf.

AEA-{O} Here A c C denotes a lattice of maximal rank. This function is (a) even: S'J( -z) = S'J(z); (b) periodic: S'J(z + -X) = S'J(z) for all -X E Aj (c) further it has no other poles than the points -X E A of the lattice. The S'J-function, being periodic, induces a well-defined meromorphic function on the elliptic curve Cj A, whose unique pole is at the origin. In particular, there is a nonconstant meromorphic function on any such curve. Lemma 1. The meromorphic functions on a Riemann surface S form a field M(S), with the natural addition and multiplication operations.

Let J: 3 1 -+ 3 2 be a nonconstant mapping of Riemann surfaces. The pull-back J*(g) of any meromorphic function 9 E M(32 ) is meromorphic on 3 1 , Example 3. Consider a holomorphic mapping J: 3 -+ CIP'2 of a Riemann surface 3 into complex projective plane C1P'2, with affine coordinates Zl, Z2. Suppose that some rational function g(Zl' Z2) is defined at least at one point of J(3). This means that it can be written as a ratio of homogeneous polynomials of the same degree: where (xo : Xl : X2) are homogeneous coordinates on CIP'2 corresponding to Zl, Z2, and the set of zeros of the polynomial q(xo, Xl, X2) does not contain J(3).