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 it's arithmetic genus <math>g</math> via the forumla:
{{subpages}}
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 formula:


<math>g=(d-1)(d-2)</math>
<math>g=\frac12 (d-1)(d-2) . \,</math>
=== proofs ===
The proof is immidiate by [[adjunction formula|adjunction]]. For a classical proof see the first reference below.


=== references ===  
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 ===
The proof follows immediately from the [[adjunction formula]]. For a classical proof see the book of Arbarello, Cornalba, Griffiths and Harris.
 
=== 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, Principls of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1
* Grffiths and Harris, Principles of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1[[Category:Suggestion Bot Tag]]

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In classical algebraic geometry, the genus-degree formula relates the degree of a non-singular plane curve with its arithmetic genus via the formula:

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

  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
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