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 stories 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's an exceptional assessment of the idea of kin among Riemann surfaces and their types - advanced algebraic curves in advanced projective areas. ... the second one paper, written through V.I.Danilov, discusses algebraic types and schemes. ...
i will suggest the e-book as an outstanding creation 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 sort is agreeable for college kids and divulges the wonderful thing about mathematics."
Acta Scientiarum Mathematicarum, 1994

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

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The de Rham isomorphism (4) carries ( , )DR into the cupproduct. This is readily seen on computing L emma. lJrt IIii = def 1 Wj an d IIig+i ai the B-periods of two closed forms Wj, 1 (Wl,W2)DR = def Wi bi via periods. are, respectwe l y, the A- and j = 1,2, then L (Ill II;+i - II;+i II;). 9 (Wl,W2)DR = i=l Let M be a development of the surface S, with symbol alblallbll ... agbga;lb;l. Then WI is exact on M, which is simply connected. We denote by 7f( q) the primitive l q Wl of Wl, where p is the starting point, and q the end point, of an integration path on M.

By additiveness, the definition of the integral of a I-form over any path reduces to the usual curvilinear integrals in the Gaussian plane C (cf. Springer [1957]). Using the following result one can define the integral of a closed I-form over an arbitrary path. Lemma. Let Ui and U2 be smooth homotopic paths on a Riemann surface S, and let w E A i be a closed form. Then By considering the pull-back of w with respect to a smooth homotopy, we reduce the proof to the corresponding fact for closed I-forms on the square [0,1] x [0,1] (see also Springer [1957]).

Every edge, ai or bi, therefore defines a loop on 8, whose homotopy class defines an element of 1f(8). Now the loop of the symbol alblal1bl1 ... agbga;lb;l is clearly homotopic to the trivial one. Thus we have defined a map, which is the required isomorphism. The proof is based on the Seifert-van Kampen theorem (see Massey [1967,1977]). Example. For an elliptic curve Cj A, the fundamental group 1f(Cj A) is isomorphic to the group with generators a, b and commutation relation aba-1b- 1 = 1. Hence it is isomorphic to the free abelian group on two generators Z EB Z (cf.

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