, By a real line in C 3 we mean a set of the form {a + tb : t ? R} where a, b ? C 3 Moreover, d (t,0,0) ?(l t ) is also a real line and it intersects d (t,0,0) ?(L k 1 ,t )?. . .?d (t,0,0) ?(L k 4 ,t ) at d (t,0,0) ?(a 1,t ),..., d (t,0,0) ?(a 4,t ) The cross-ratio of the last four points equals that of a 1,t, d (t,0,0) ?(a 4,t ) equals the cross-ratio of d (t,0,0) ?(L k 1 ,t ),..., d (t,0,0) ?(L k 4 ,t ), we obtain our claim. Now observe that the complex t-axis is the singular locus of V and its image S by ? is the singular locus of ?(V ). Clearly, S is a smooth complex Nash curve,t ), respectively, depend algebraically on s = ?(t, 0, 0) ? S (cf + t and ?(t), respectively. The last two paragraphs imply that for t ? R with |t| small)) = ?(t) Since S is a smooth complex Nash curve, we may assume that h 1 , h 2 are defined in some, Moreover, the crossratios h 1 , h 2 of d (t,0,0) ?(L 1,t ), . . . , d (t,0,0) ?(L 4,t ) and d (t,0,0) ?(L 1 : S ? C are complex Nash functions. On the other hand We have ?(t) = ? 1 (t) + i? 2 (t) where ? 1 , ? 2 are real valued continuous functions and h 1 (s) = u 1 (s) + iv 1 (s), where u 1 , v 1 are real Nash functions)) = 0 for t ? R with |t| small. Since u 1 , v 1 satisfy the Cauchy-Riemann equations and h 1 is not constant, neither of u 1 , v 1 is constant. Consequently, ? 1 | R , ? 2 | R are semi-algebraic functions, which contradicts the fact that h 2 (?(t)) = ?(t) for real t
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