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 μ₂ : J [QF] → ℍ² × ℍ². To this end, we first make a careful study of the algebraic curves in the algebraic surface Φ ≅ {(x, y, z) ∈ ℂ³ I x² + y² + z² = 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⁺) ∈ ℍ² × ℍ² (Lemmas 9.1.8 and 9.1.12). We also observe that the number of the algebraic roots for ν 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 geometric continuity. By using the idea, we show that all geometric roots for a given label ν 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” d _G (ν) of a label ν, and then show that d _G (ν) = 1 for every ν by using the argument of geometric continuity (Proposition 9.2.3).