Genus–degree formula

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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[edit]

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

Generalization[edit]

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[edit]

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

References[edit]