Genus-degree formula: Difference between revisions
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imported>David E. Volk m (subpages) |
imported>Richard Pinch (corrected formula, added ref to effect of singular points) |
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In classical [[algebraic geometry]], the genus-degree formula relates the degree <math>d</math> of a plane curve <math>C\subset\mathbb{P}^2</math> with its arithmetic genus <math>g</math> via the forumla: | 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 (geometry)|genus]] <math>g</math> via the forumla: | ||
<math>g=\frac12 (d-1)(d-2) . \,</math> | |||
A [[singularity]] of order ''r'' decreases the genus by <math>\scriptstyle \frac12 r(r-1)</math>.<ref>Semple and Roth, ''Introduction to Algebraic Geometry'', Oxford University Press (repr.1985) ISBN 0-19-85336-2. Pp.53-54</ref> | |||
=== Proofs === | === Proofs === | ||
The proof | The proof follows immediately from the [[adjunction formula]]. For a classical proof see the book of Arbarello, Cornalba, Griffiths and Harris. | ||
=== References === | === References === | ||
{{reflist}} | |||
* Arbarello, Cornalba, Griffiths, Harris. Geometry of algebraic curves. vol 1 Springer, ISBN 0387909974, appendix A. | * Arbarello, Cornalba, Griffiths, Harris. Geometry of algebraic curves. vol 1 Springer, ISBN 0387909974, appendix A. | ||
* Grffiths and Harris, | * Grffiths and Harris, Principles of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1 | ||
Revision as of 05:28, 26 October 2008
In classical algebraic geometry, the genus-degree formula relates the degree of a non-singular plane curve with its arithmetic genus via the forumla:
A singularity of order r decreases the genus by .[1]
Proofs
The proof follows immediately from the adjunction formula. For a classical proof see the book of Arbarello, Cornalba, Griffiths and Harris.
References
- ↑ 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