In classicalalgebraic geometry, thegenus–degree formula relates the degree${\displaystyle d}$ of anirreducibleplane curve${\displaystyle C}$ with itsarithmetic 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 theprojective plane${\displaystyle {\mathbb{P}}^2}$. If the curve is non-singular thegeometric genus and thearithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinarysingularity of multiplicity${\displaystyle r}$ decreases the genus by${\displaystyle \frac{1}{2}r(r-1)}$.^[1]

Elliptic curves are parametrized byWeierstrass elliptic functions. Since these parameterizing functions are doubly periodic, the elliptic curve can be identified with a period parallelogram with the sides glued togetheri.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$${\displaystyle \left(y^2-x(x-1)(x-2)\right)(5y-x)+ϵx=0.}$$For small nonzero${\displaystyle \epsilon }$ this is gives the nonsingular curve. However, when${\displaystyle \epsilon =0}$, 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${\displaystyle \phi }$ 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${\displaystyle \epsilon }$-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$${\displaystyle g(d+1)=g(d)+d-1,g(1)=0.}$$This recursion relation has the solution${\displaystyle g(d)=\frac{1}{2}(d-1)(d-2)}$.

The genus–degree formula can be proven from theadjunction formula; for details, seeAdjunction formula § Applications to curves.^[2]

For a non-singularhypersurface${\displaystyle H}$ of degree${\displaystyle d}$ in theprojective space${\displaystyle {\mathbb{P}}^n}$ ofarithmetic genus${\displaystyle g}$ the formula becomes:

${\displaystyle g=(\genfrac{}{}{0ex}{}{d-1}{n}),}$

where${\displaystyle (\genfrac{}{}{0ex}{}{d-1}{n})}$ is thebinomial coefficient.

• Thom conjecture

• This article incorporates material from theCitizendium article "Genus degree formula", which is licensed under theCreative Commons Attribution-ShareAlike 3.0 Unported License but not under theGFDL.
• Enrico Arbarello, Maurizio Cornalba,Phillip Griffiths,Joe Harris. Geometry of algebraic curves. vol 1 Springer,ISBN 0-387-90997-4, appendix A.
• Phillip Griffiths andJoe 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

• Algebraic curves

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