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 genusg 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 genusg 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