In classical algebraic geometry, the genus–degree formula relates the degree ''d'' of an irreducible plane curve

C

with its arithmetic genus ''g'' via the formula: :

g=\frac12 (d-1)(d-2).

Here "plane curve" means that

C

is a closed curve in the

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...

\mathbb^2

. If the curve is non-singular the

geometric genus In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex ...

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

\frac12 r(r-1)

Proof

The proof follows immediately from the

adjunction formula In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedde ...

. For a classical proof see the book of Arbarello, Cornalba, Griffiths and Harris.

Generalization

For a non-singular

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

H

of degree ''d'' in the projective space

\mathbb^n

of arithmetic genus ''g'' the formula becomes: :

g=\binom , \,

where

\tbinom

is the binomial coefficient.

Notes

References

* *

Enrico Arbarello Enrico Arbarello is an Italian mathematician who is a leading expert in algebraic geometry. He earned a Ph.D. at Columbia University in New York in 1973. He was a visiting scholar at the Institute for Advanced Study from 1993-94. He is now a M ...

, Maurizio Cornalba,

Phillip Griffiths Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particul ...

, Joe Harris. Geometry of algebraic curves. vol 1 Springer, , appendix A. *

and Joe Harris, Principles of algebraic geometry, Wiley, , chapter 2, section 1. *

Robin Hartshorne __NOTOC__ Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry. Career Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...

(1977): ''Algebraic geometry'', Springer, . * {{DEFAULTSORT:Genus-degree formula Algebraic curves

Proof

Generalization

Notes

See also

References