# Genus–degree formula

(Redirected from Degree-genus 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:

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

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

## Proof

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

## Generalization

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

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

where $\tbinom{d-1}{n}$ 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