# Genus–degree formula

(Redirected from Degree-genus formula)

In classical algebraic geometry, the genus–degree formula relates the degree d of an irreducible plane curve ${\displaystyle C}$ with its arithmetic genus g via the formula:

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

Here "plane curve" means that ${\displaystyle C}$ is a closed curve in the projective plane ${\displaystyle \mathbb {P} ^{2}}$. 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 ${\displaystyle \scriptstyle {\frac {1}{2}}r(r-1)}$.[1]

## Proof

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

## Generalization

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

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

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

## Notes

1. ^ Semple and Roth, Introduction to Algebraic Geometry, Oxford University Press (repr.1985) ISBN 0-19-853363-2. Pp. 53–54