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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 (() ()) + =
The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.
On a general Riemann surface of genus , has degree , independently of the meromorphic form chosen to represent the divisor. This follows from putting D = K {\displaystyle D=K} in the theorem. In particular, as long as D {\displaystyle D} has degree at least 2 g − 1 {\displaystyle 2g-1} , the correction term is 0, so that
Using the Riemann–Hurwitz formula, the hyperelliptic curve with genus g is defined by an equation with degree n = 2g + 2. Suppose f : X → P 1 is a branched covering with ramification degree 2, where X is a curve with genus g and P 1 is the Riemann sphere. Let g 1 = g and g 0 be the genus of P 1 ( = 0 ), then the Riemann-Hurwitz formula ...
The formula may also be used to calculate the genus of hyperelliptic curves. As another example, the Riemann sphere maps to itself by the function z n, which has ramification index n at 0, for any integer n > 1. There can only be other ramification at the point at infinity. In order to balance the equation
However, some properties are not kept under birational equivalence and must be studied on non-plane curves. This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula).
A real hyperelliptic curve of genus g over K is defined by an equation of the form : + = where () has degree not larger than g+1 while () must have degree 2g+1 or 2g+2. This curve is a non singular curve where no point (,) in the algebraic closure of satisfies the curve equation + = and both partial derivative equations: + = and ′ = ′ ().
For genus g ≥ 3 it is no longer the case that the genus determines the gonality. The gonality of the generic curve of genus g is the floor function of (g + 3)/2. Trigonal curves are those with gonality 3, and this case gave rise to the name in general. Trigonal curves include the Picard curves, of genus three and given by an equation y 3 = Q(x)