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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 (() ()) + =
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)
1.1.7 Families of variable degree. 1.2 Curves of genus one. ... Curves with genus greater than one. Butterfly curve (algebraic) Elkies trinomial curves.
Rational curves are subdivided according to the degree of the polynomial. Degree 1 Line ... Bolza surface (genus 2) Klein quartic (genus 3) Bring's curve (genus 4)
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 ′ = ′ ().
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 curve has genus one (genus formula); in particular, it is not isomorphic to the projective line P 1, which has genus zero. Using genus to distinguish curves is very basic: in fact, the genus is the first invariant one uses to classify curves (see also the construction of moduli of algebraic curves).