Genus–degree formula explained

C with its arithmetic genus g via the formula:g = (12)/((d -1 )(d -2 ).) Here "plane curve" means that C is a closed curve in the projective plane P ^{2} . If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary singularity of multiplicity r decreases the genus by (12)/(r(r -1 )) .^[1]

Proof

The genus–degree formula can be proven from the adjunction formula; for details, see .^[2]

Generalization

H of degree d in the projective space P ^{n} of arithmetic genus g the formula becomes:g = \binom{d -1 }{n} ,

where

\tbinom{d -1 }{n} is the binomial coefficient.

Notes

1. Book: Semple. John Greenlees Semple. John Greenlees. Roth. Leonard . Leonard Roth. Introduction to Algebraic Geometry. . 1985 . 0814690 . 0-19-853363-2. 53–54.
2. Algebraic geometry, Robin Hartshorne, Springer GTM 52,, chapter V, example 1.5.1

See also

• Thom conjecture

References

• • Enrico Arbarello, Maurizio Cornalba, Phillip Griffiths, Joe Harris. Geometry of algebraic curves. vol 1 Springer,, appendix A.
• Phillip Griffiths and Joe Harris, Principles of algebraic geometry, Wiley,, chapter 2, section 1.
• Robin Hartshorne (1977): Algebraic geometry, Springer, .