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462 PHILLIP A. GRIFFITHS
de1 o)11 el (-012 e2
de2 o)21 el 0922 e2 -I- 092a e3.
As before, from o h21a091 -t- h22a092 and h21a h123 0 we have oa t9 oz.
From exterior differentiation of 0)13 0,
0 &o 13 (.012/ 0)23
which implies that o o-oz. The fibres of 3’ are then defined by (02 0, and
along these curves dz o)1 el, de o11 e as before.
Thus in both cases the fibres of the Gauss map are lines, and consequently S
is a ruled surface. As such, it may be given parametrically by
(t, x) v(t) + Xw(t)
where v(t), w(t) are holomorphic vector-valued functions and h is a linear parameter
on the line Lt. This representation is unique up to a substitution
as depicted by
The Gauss map is
f(t) v(t) + h(t)w(t)
(t) (t) w(t)
y(t, X) (v’ + hw’)/k w
and this map is degenerate exactly when v’/ w and w’/ w are proportional.
If w’/ w 0 so that w’ cw, then i,’ ’w + gaw and we may make ri,’ 0
by solving ’ + c 0. In this case S is a cone (including the special case of a
cylinder). If w’/ w 0, then
v’ / w r w’ / w.
Under the above substitution (and taking # w),