In classical algebraic geometry, the genus–degree formula relates the degree
of an irreducible plane curve
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
[1] 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.
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.
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.
is the genus of a curve of degree
nonsingular curve, then proceeding as above, we obtain a nonsingular curve of degree
-smoothing the union of a curve of degree
The line meets the degree
points, so this leads to an recursion relation
This recursion relation has the solution