en>Linas: clarify, make it simpler

2012-07-29T16:09:23Z

clarify, make it simpler

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

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

A [[Mathematical singularity|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>

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

== Generalization ==
For a non-singular [[hypersurface]] <math>H</math> of degree ''d'' in <math>\mathbb{P}^n</math> of [[arithmetic genus]] ''g'' the formula becomes:

: <math>g=\binom{d-1}{n} , \,</math>

where <math>\tbinom{d-1}{n}</math> is the [[binomial coefficient]].

==Notes==
<references/>

== References ==
*{{citizendium|title=Genus degree formula}}
* Arbarello, Cornalba, Griffiths, Harris. Geometry of algebraic curves. vol 1 Springer, ISBN 0-387-90997-4, appendix A.
* Grffiths and Harris, Principles of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1
*{{springer | title=Genus of a curve | id=G/g043990 | last=Kulikov | first=Viktor S. }}

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en>Linas: clarify, make it simpler