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 by way of V.V.Shokurov, is dedicated to the idea of Riemann surfaces and algebraic curves. it's a good evaluate of the idea of kin among Riemann surfaces and their types - advanced algebraic curves in advanced projective areas. ... the second one paper, written via V.I.Danilov, discusses algebraic forms and schemes. ...

i will suggest the booklet as a great advent to the fundamental algebraic geometry."

European Mathematical Society e-newsletter, 1996

"... To sum up, this publication is helping to profit algebraic geometry very quickly, its concrete type is pleasant for college students and divulges the wonderful thing about mathematics."

Acta Scientiarum Mathematicarum, 1994

**Read Online or Download Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes PDF**

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**Extra info for Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes**

**Sample text**

8 and cf. Sect. 7 of Chap. 2). Let F(xo, Xl, X2) be an irreducible homogeneous polynomial defining C. , X2), X2 - 1) is the ideal generated by the polynomials F(xo, Xl, X2) and X2 - 1. The mapping f(p) = (xo(p) : XI(P) : 1), where pES, extends by continuity to a desingularization of the curve C. As a special case, if C is nonsingular then 8 is the Riemann surface of C and f = id. If, on the other hand, C has some singularities then f resolves them, for it is an isomorphism over the nonsingular points.

Further F satisfies equation (2). By Riemann's removable singularity theorem, the coefficients Ci have meromorphic continuations on S2' We return to the construction of the Riemann surface of the algebraic function F. Note that this function is holomorphic on U and is given by the projection map (p, z) 1---+ z. 1. Again, by Riemann's removable singularity theorem and by the Proposition of Sect. 10, we obtain the required mapping f: Sl --+ S2, U eSt, and a meromorphic function F on Sl. But according to Proposition 2 the decomposition of Sl into connected components defines a factorization of P, whence Sl (and U) are connected.

Definition. A mapping of Riemann surfaces f: 8 1 mal, or Galois, if its automorphism group --t 8 2 is said to be nor- Aut f ~f {g E Aut 8 1 I fog = J} acts transitively on the fibres f- 1 (p), p E 8 2 . Corollary 2. 1)jF. 11. The Riemann Surface of an Algebraic Function Proposition 1. Let be an irreducible polynomial. Then there exists a finite mapping of Riemann surfaces f: 8 1 --t 8 2 , of degree n, and a meromorphic function F E M(81 ) that satisfies the equation Fn + r(Cl) F n - l + ... + r(en) = O.