Colloquium Lectures Volume 4

Colloquium Lectures Volume 4

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1914 edition. Excerpt: ...and the Abelian integrals as developed by Riemann, early made itself felt in the study of algebraic surfaces and algebraic functions of two variables. Thus we find a paper by ClebschJ of the year 1868, in which he discovers an invariant of an algebraic surface analogous to the deficiency p of an algebraic curve. The latter invariant may be defined as the number of essential constants in the general integral of the first kind, i. e., in the everywhere finite integral, and this integral can be written in the form Cf. preceding reference. Furthermore Hecke, Gottinger Dissertation, 1910. t Introduzione alia teoria dei gruppi discontinui e delle funzioni automorfe, 1908. t C. R., 67 (1868), p. 1238. Clebsch had only the adjoint Cs of degree m--4. The everywhere finite double integral is due to Noether, Math. An., 2 (1870), p. 293. where /(, y) 0 is the equation of the ground curve Cm, assumed irreducible, and Q(x, y) is an adjoint polynomial of degree m--3. If, in particular, Cm has only ordinary double points, Q = 0 is any that passes through these points. Consider now an irreducible algebraic surface f(x, y, z) = 0 of degree m with only ordinary multiple lines and isolated multiple points. Then the double integral (II, 2) taken over an arbitrary regular surface, open or closed, lying in the four-dimensional Riemann manifold corresponding to the function z of x, y defined by the equation / = 0, will remain finite provided Qx, y, z) = 0 is an adjoint surface of degree in--4, i. e., a surface which passes through the multiple lines and has a multiple line of order Jb--1 at least in every multiple line of /of order k; and which moreover has a multiple point of order q--2 at least in every isolated multiple point of / of order q. Such...
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Product details

  • Paperback
  • 189 x 246 x 3mm | 113g
  • United States
  • English
  • black & white illustrations
  • 1236875680
  • 9781236875686