Geometry

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... 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

Example text

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 [1986]), 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.

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