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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