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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 (() ()) + =
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 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.
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)
Conversely, any smooth closed curve in of degree three has genus one by the genus formula and is thus an elliptic curve. A smooth complete curve of genus greater than or equal to two is called a hyperelliptic curve if there is a finite morphism C → P 1 {\displaystyle C\to \mathbb {P} ^{1}} of degree two.
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).
The first two invariants covered by the Plücker formulas are the degree d of the curve C and the degree d *, classically called the class of C. Geometrically, d is the number of times a given line intersects C with multiplicities properly counted. (This includes complex points and points at infinity since the curves are taken to be subsets of ...