Genus–degree formula

In classical algebraic geometry, the genus–degree formula relates the degree ${\displaystyle d}$ of an irreducible plane curve ${\displaystyle C}$ with its arithmetic genus ${\displaystyle g}$ via the formula:

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

Here "plane curve" means that ${\displaystyle C}$ is a closed curve in the projective plane ${\displaystyle \mathbb {P} ^{2}}$. 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 ${\displaystyle r}$ decreases the genus by ${\displaystyle {\frac {1}{2}}r(r-1)}$.^[1]

Motivation

Elliptic curves are parametrized by Weierstrass elliptic functions. Since these parameterizing functions are doubly periodic, the elliptic curve can be identified with a period parallelogram with the sides glued together i.e. a torus. So the genus of an elliptic curve is 1. The genus–degree formula is a generalization of this fact to higher genus curves. The basic idea would be to use higher degree equations. Consider the quartic equation $${\displaystyle \left(y^{2}-x(x-1)(x-2)\right)\,(5y-x)+\epsilon x=0.}$$ For small nonzero ${\displaystyle \varepsilon }$ this is gives the nonsingular curve. However, when ${\displaystyle \varepsilon =0}$, this is $${\displaystyle \left(y^{2}-x(x-1)(x-2)\right)\,(5y-x)=0,}$$ a reducible curve (the union of a nonsingular cubic and a line). When the points of infinity are added, we get a line meeting the cubic in 3 points. The complex picture of this reducible curve looks like a torus and a sphere touching at 3 points. As ${\displaystyle \varphi }$ changes to nonzero values, the points of contact open up into tubes connecting the torus and sphere, adding 2 handles to the torus, resulting in a genus 3 curve.

In general, if ${\displaystyle g(d)}$ is the genus of a curve of degree ${\displaystyle d}$ nonsingular curve, then proceeding as above, we obtain a nonsingular curve of degree ${\displaystyle d+1}$ by ${\displaystyle \varepsilon }$-smoothing the union of a curve of degree ${\displaystyle d}$ and a line. The line meets the degree ${\displaystyle d}$ curve in ${\displaystyle d}$ points, so this leads to an recursion relation $${\displaystyle g(d+1)=g(d)+d-1,\quad g(1)=0.}$$ This recursion relation has the solution ${\displaystyle g(d)={\frac {1}{2}}(d-1)(d-2)}$.

Proof

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

Generalization

For a non-singular hypersurface ${\displaystyle H}$ of degree ${\displaystyle d}$ in the projective space ${\displaystyle \mathbb {P} ^{n}}$ of arithmetic genus ${\displaystyle g}$ the formula becomes:

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

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

Notes

1. Semple, John Greenlees; Roth, Leonard. Introduction to Algebraic Geometry (1985 ed.). Oxford University Press. pp. 53–54. ISBN 0-19-853363-2. MR 0814690.
2. Algebraic geometry, Robin Hartshorne, Springer GTM 52, ISBN 0-387-90244-9, chapter V, example 1.5.1

See also

• Thom conjecture

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.
• Enrico Arbarello, Maurizio Cornalba, Phillip Griffiths, Joe Harris. Geometry of algebraic curves. vol 1 Springer, ISBN 0-387-90997-4, appendix A.
• Phillip Griffiths and Joe 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", Encyclopedia of Mathematics, EMS Press