Genus–degree formula

In classical algebraic geometry, the genus–degree formula relates the degree d of an irreducible plane curve $$C$$ with its arithmetic genus g via the formula:
 * $$g=\frac12 (d-1)(d-2).$$

Here "plane curve" means that $$C$$ is a closed curve in the projective plane $$\mathbb{P}^2$$. If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary singularity of multiplicity r decreases the genus by $$\frac12 r(r-1)$$.

Proof
The genus–degree formula can be proven from the adjunction formula; for details, see.

Generalization
For a non-singular hypersurface $$H$$ of degree d in the projective space $$\mathbb{P}^n$$ of arithmetic genus g the formula becomes:


 * $$g=\binom{d-1}{n}, \,$$

where $$\tbinom{d-1}{n}$$ is the binomial coefficient.