Genus-degree formula: Difference between revisions
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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 | 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: | ||
<math>g=(d-1)(d-2)</math> | <math>g=(d-1)(d-2)</math> | ||
=== | === Proofs === | ||
The proof is | The proof is immediate by [[adjunction formula|adjunction]]. For a classical proof see the first reference below. | ||
=== | === References === | ||
* 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, Principls of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1 | * Grffiths and Harris, Principls of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1 | ||
Revision as of 08:16, 21 March 2008
In classical algebraic geometry, the genus-degree formula relates the degree of a plane curve with its arithmetic genus via the forumla:
Proofs
The proof is immediate by adjunction. For a classical proof see the first reference below.
References
- Arbarello, Cornalba, Griffiths, Harris. Geometry of algebraic curves. vol 1 Springer, ISBN 0387909974, appendix A.
- Grffiths and Harris, Principls of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1