Genus–degree formula

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={\frac {1}{2}}(d-1)(d-2).

Here "plane curve" means that $C$ is a closed curve in the projective plane $\mathbb {P} ^{2}$ . 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 $\scriptstyle {\frac 12}r(r-1)$ .^[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-singular hypersurface $H$ of degree d in the projective space $\mathbb {P} ^{n}$ of arithmetic genus g the formula becomes:

g={\binom {d-1}{n}},\,

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

Notes

↑ Semple and Roth, Introduction to Algebraic Geometry, Oxford University Press (repr.1985) ISBN 0-19-853363-2. Pp. 53–54

References

This article incorporates material from the Citizendium article "Genus degree formula", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.
Arbarello, Cornalba, Griffiths, Harris. Geometry of algebraic curves. vol 1 Springer, ISBN 0-387-90997-4, appendix A.
Griffiths and Harris, Principles of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1.
Robin Hartshorne (1977): Algebraic geometry, Springer, ISBN 0-387-90244-9.
Kulikov, Viktor S. (2001) [1994], "Genus of a curve", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

Topics in algebraic curves

Rational curves

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Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

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Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem
Singularities	Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

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