Genus-degree formula

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In classical algebraic geometry, the genus-degree formula relates the degree <math>d</math> of a non-singular plane curve <math>C\subset\mathbb{P}^2</math> with its arithmetic genus <math>g</math> via the formula:

<math>g=\frac12 (d-1)(d-2) . \,</math>

A singularity of order r decreases the genus by <math>\scriptstyle \frac12 r(r-1)</math>.[1]

Proofs

The proof follows immediately from the adjunction formula. For a classical proof see the book of Arbarello, Cornalba, Griffiths and Harris.

References

  1. Semple and Roth, Introduction to Algebraic Geometry, Oxford University Press (repr.1985) ISBN 0-19-85336-2. Pp.53-54
  • Arbarello, Cornalba, Griffiths, Harris. Geometry of algebraic curves. vol 1 Springer, ISBN 0387909974, appendix A.
  • Grffiths and Harris, Principles of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1