Continuity set

In classical algebraic geometry, the genus–degree formula relates the degree d of a non-singular plane curve $C \subset ℙ^{2}$ 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 $\frac{1}{2} r (r - 1)$ .^[1]

Proof

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

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

g = (\binom{d - 1}{n}),

↑ Semple and Roth, Introduction to Algebraic Geometry, Oxford University Press (repr.1985) ISBN 0-19-85336-2. Pp. 53–54

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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