Genus-degree formula

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:

$g={\frac {1}{2}}(d-1)(d-2).\,$

A singularity of order r decreases the genus by $\scriptstyle {\frac {1}{2}}r(r-1)$.

Proofs

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

References

• 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