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Global Investigations

In: Plane Algebraic Curves

Author

Listed:
  • Egbert Brieskorn

    (Universität Bonn, Mathematisches Institut)

  • Horst Knörrer

    (Universität Bonn, Mathematisches Institut)

Abstract

Now that we have investigated the local properties of plane curves and their singular points from various viewpoints, in the previous paragraph, we next want to derive global assertions about plane complex projective algebraic curves. Above all, we shall calculate global invariants of such curves from the local invariants of their singular points which we have investigated earlier. The global invariants in question are the order, class and genus of curves. Formulae for the order and class were first presented by J. Plücker in 1834, and later generalised by M. Noether 1875 and 1883. We want to deal with these formulae in the present section. Then in the next section we shall derive the formula for the genus, which in special cases goes back to Riemann 1857 and Clebsch 1864, and in its general form to M. Noether 1874 and Weierstrass.

Suggested Citation

  • Egbert Brieskorn & Horst Knörrer, 1986. "Global Investigations," Springer Books, in: Plane Algebraic Curves, chapter 9, pages 576-693, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-5097-1_9
    DOI: 10.1007/978-3-0348-5097-1_9
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