Genus-degree formula

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

$g=(d-1)(d-2)$

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