Genus–degree formula explained

In classical algebraic geometry, the genus–degree formula relates the degree

of an irreducible plane curve with its arithmetic genus via the formula: Here "plane curve" means that is a closed curve in the projective plane . 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 decreases the genus by .^[1]

Proof

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

Generalization

of degree in the projective space of arithmetic genus the formula becomes:

where

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, .