Continuity set

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 $\scriptstyle {\frac {1}{2}}r(r-1)$.

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.

References

• Template:Citizendium
• Arbarello, Cornalba, Griffiths, Harris. Geometry of algebraic curves. vol 1 Springer, ISBN 0-387-90997-4, appendix A.
• Grffiths and Harris, Principles of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1
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