12 Takashi SfflOYA By eliminating lemons one gets the following: 1.11. Proposition. Let a be an almost regular curve. (i) If ind(a) = °°, then rot(a) = «. (ii) If ind(a) °°, then rot(a) = ind(a) - 1 if ~n{a) - ind(a) is odd, and rot(a) = ind(a) ifn(cc) - ind(a) is even or infinite. • Corollary. A semi-regular curve a is regular if and only if rot(a) = ind(a) = n(cc). D 1.12. Proposition. Let a be a semi-regular curve and K a closed disk such that K n a = 0 and that, when a is not simple, K C Int(Bi). Let N := C1(M - K) and let A ^ be the universal covering of N. Then any lifting a of a is a semi- regular curve in the differentiable plane N - dN. Moreover if a is almost regular, the lifting a is simple. • Exercise. For a general semi-regular curve a characterize the set of integers i e \n(a)] such that the inverse image of the double point pt e a i s a double point of a, and give necessary and sufficient conditions on the sequence {&(/)} for & to be simple (resp. regular, almost regular).

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