# Algebraic roots and geometric roots

The purpose of this chapter is to prove Proposition 6.2.6 (Unique realization), which implies the bijectivity of the map *μ*_{2} : *J* [*QF*] → ℍ^{2} × ℍ^{2}. To this end, we first make a careful study of the algebraic curves in the algebraic surface Φ ≅ {(*x*, *y*, *z*) ∈ ℂ^{3} I *x*^{2} + *y*^{2} + *z*^{2} = *xyz*} determined by the equations in Definition 4.2.19, and find the irreducible components which contain the *geometric roots* (Definition 9.1.2) for a given label * ν* = (

*ν*

^{−},

*ν*

^{+}) ∈ ℍ

^{2}× ℍ

^{2}(Lemmas 9.1.8 and 9.1.12). We also observe that the number of the algebraic roots for

*is finite (Proposition 9.1.13). Thus the problem is how to single out the geometric roots among the algebraic roots. Our answer is to appeal to the idea of the*

**ν***geometric continuity*. By using the idea, we show that all geometric roots for a given label

*are obtained by continuous deformation of the unique geometric root for a special label corresponding to a fuchsian group. This implies the desired result that each label has the unique geometric root. To realize this idea, we introduce the concept of the “geometric degree”*

**ν***d*

_{ G }(

*) of a label*

**ν***, and then show that*

**ν***d*

_{ G }(

*) = 1 for every*

**ν***by using the argument of geometric continuity (Proposition 9.2.3).*

**ν**## Keywords

Irreducible Component Algebraic Variety Intersection Number Fuchsian Group Smooth Point## Preview

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