# Genus–degree formula

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# Genus–degree formula

Genus–degree formula

`In classicalalgebraic geometry, the`

**genus–degree formula**relates the degree*d*of an irreducible plane curvewith itsarithmetic genus*g*via the formula:`Here "plane curve" means thatis a closed curve in theprojective plane. If the curve is non-singular thegeometric genusand thearithmetic genusare equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinarysingularityof multiplicity`

*r*decreases the genus by.^{[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-singularhypersurfaceof degree`

*d*in theprojective spaceofarithmetic genus*g*the formula becomes:`whereis thebinomial coefficient.`

## References

[1]

Citation Link//www.ams.org/mathscinet-getitem?mr=0814690Semple, John Greenlees; Roth, Leonard.

*Introduction to Algebraic Geometry*(1985 ed.). Oxford University Press. pp. 53–54. ISBN 0-19-853363-2. MR 0814690.Sep 24, 2019, 9:09 PM

[5]

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Sep 24, 2019, 9:09 PM