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... includes papers. the 1st, written by means of V.V.Shokurov, is dedicated to the idea of Riemann surfaces and algebraic curves. it's a great evaluation of the speculation of kin among Riemann surfaces and their types - complicated algebraic curves in complicated projective areas. ... the second one paper, written by means of V.I.Danilov, discusses algebraic forms and schemes. ...
i will be able to suggest the e-book as an outstanding creation to the elemental algebraic geometry."
European Mathematical Society publication, 1996
"... To sum up, this e-book is helping to benefit algebraic geometry very quickly, its concrete kind is pleasing for college students and divulges the great thing about mathematics."
Acta Scientiarum Mathematicarum, 1994
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Additional info for Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes
The differentiations at a point p form a complex vector space with the natural operations of addition and multiplication by constants. This vector space, denoted by Tp(8), is called the tangent space to 8 at p. 1. Riemann Surfaces and Algebraic Curves 45 = x + Ay be a local coordinate at a point p E 8. Then the partial derivatives, ~~ (p) and ~; (p), of the functions f E £(8), written Examples. Let z in the local coordinates xand y, determine differentiations :x \ and :y \ p p at p. Further examples of differentiations are provided by the operators of Wirtinger's calculus: We observe that a holomorphic function on an open set U C C is nothing but a differentiable function f E £(U) that satisfies the Cauchy-Riemann equation :zf = 0 (cf.
It would be more convenient to present the hyperelliptic surface S as being the plane curve y2 = J(z). But, for n ::::: 4 this has a singular point at infinity. Hence S may be viewed as its desingularization (see Corollary 4 below). In view of the primitive element theorem (cf. Shafarevich ), we obtain from the above Propositions: Theorem 1. If Sl field extension --+ S2 is a finite mapping of Riemann surfaces then the '-+ M(Sl) is finite, and its degree is ::; degf. 1*: M(S2) Theorem 2. Let S2 be a Riemann surface and let cp: M(S2) '-+ K be a finite C-extension of degree n.
Theorem. A compact Riemann surface S has a development with symbol (1) aa-l, or b -lb- l (2) al bla -lb-l I ... a g gag g . l Corollary. In case (1), the Riemann surface S is homeomorphic to a sphere; in case (2), to a sphere with g handles. 38 V. V. Shokurov Thus we see (but we have not proved) that the symbols in the Theorem are topological invariants of the Riemann surface. Definition. The number 9 in (2) (and 0, in case (1)) is called the (topological) genus of the compact Riemann surface 8.