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5. + − (Nd,e , respectively) consists of all de Jonqi`eres (a) The group Nd,e transformations ϕ+ as in (7) (ϕ− as in (8), respectively) with f ∈ Ad,e (f ∈ Ad,e , respectively). ± (b) The subgroup Nd,e is the centralizer of Gd,e in the group ± Jonq (A2 ). Proof. We stick to the plus-case, the proof in the other one being similar. We have ϕ+ ◦ g ◦ (ϕ+ )−1 : (x, y) −→ y y ζ e x + f (ζ ) − ζ e f ( ), ζy β β . Hence ϕ+ ◦ g ◦ (ϕ+ )−1 ∈ Gd,e if and only if f (ζt) = ζ e f (t), if and only if f ∈ Ad,e .

3 a suitable automorphism γ ∈ Aut(A2 ) sends the reduced curve C = π∗ (C) to a union C = C 1 + . . + C r of affine lines through the origin given by equation (16) y(y − κ2 x) . . (y − κr x) = 0, where κi ∈ C× are distinct . The curve C is stable under the action on A2 of the cyclic group γGd,e γ −1 = g , where g = γgγ −1 ∈ Aut(A2 ). 9 Stab(C ) ⊆ GL(2, C), hence g ∈ GL(2, C). There exists an element δ ∈ GL(2, C) such that δg δ −1 = g is diagonal and acts via (x, y) → (ζ e x, ζy). Since no component C i of C is stable under g, the composition δγ sends each C i to a line through the origin different from a coordinate axis.

Clearly Ta,b ⊆ Stab(C). 12 Stab(C) = Ta,b . Suppose that r > 1. Consider a finite abelian group H = Stab(C) ∩ T1,0 . We claim that Stab(C) = H · Ta,b ⊆ T. Hence this is a quasitorus of rank one, as stated. Indeed, if δ ∈ Stab(C) \ Ta,b then δ(C 1 ) = C i for some i > 1. If h ∈ T1,0 is such that h(C i ) = C 1 then γ = h ◦ δ ∈ Stab(C 1 ) = Ta,b . Hence h = γ ◦ δ −1 ∈ H and so δ = h−1 ◦ γ ∈ H · Ta,b . Now the claim follows. This ends the proof. 14. Let C be an acyclic curve C of type (V) given by equation (2), where r ≥ 1 and εy + r ≥ 2.