By Dolgachev I.V.

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**Sample text**

This gives a construction of an automorphism gp0 associated to a double points p0 . One can do it in formulas. Without loss of generality, we may assume that p0 has the coordinates [1, 0, 0, 0], so that we can write the equation of X in the form t20 Q(t1 , t2 , t3 ) + 2t0 Φ3 (t1 , t2 , t3 ) + P hi4 (t1 , t2 , t3 ) = 0. 3) The line passing through the point x0 has a parameter equation [s, ta1 , ta2 , ta3 ]. Plugging this in the equation and cancelling by t2 , we obtain s2 Q(a1 , a2 , a3 ) + 2stΦ3 (a1 , a2 , a3 ) + t2 Φ4 (a1 , a2 , a3 ) = 0.

4) defined by [s2 , t2 ] = [−Φ4 (a1 , a2 , a3 ), Φ3 (a1 , a2 , a3 )]. The only case where this does not make sense is when Q(a1 , a2 , a3 ) = Φ4 (a1 , a2 , a3 ) = Φ3 (a1 , a2 , a3 ) = 0. This happens if and only if the line is contained in the surface. One can show that the map T : x → x still extends to X and fixes any point on this line including the point (a1 , a2 , a3 ) lying on the exceptional curve of σ : X → X . Other fixed points of the automorphism gp0 : X → X lie on the line with the slope [a1 , a2 , a3 ] satisfying the discriminant equation of degree 6 D(t1 , t2 , t3 ) = Φ3 (t1 , t2 , t3 )2 − Q(t1 , t2 , t3 )Φ4 (t1 , t2 , t3 ) = 0.

In terminology due to P. Sarnak, Γ is a thin group. 6. Let Γ be an integral Kleinian group of isometries of Hn of finite covolume, for example, the orthogonal group of some integral quadratic lattice L. It is obviously geometrically finite, however, for n > 3 it may contain finitely generated subgroups which are not geometrically finite. In fact, for any lattice L of rank ≥ 5 that contains a primitive sublattice I 1,3 , the orthogonal group O(L) contains finitely generated but not finitely presented subgroups (see [31]).

### Hyperbolic geometry and Algebraic geometry by Dolgachev I.V.

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