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# Genus-degree formula

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

$g=\frac12 (d-1)(d-2) . \,$

A singularity of order r decreases the genus by $\scriptstyle \frac12 r(r-1)$.[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