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498 PHILLIP A. GRIFFITHS
where the vanishing cycle 8t Hl(Wt) shrinks to a point as O, we have
x(V,) X(Vo) 2, t o.
It follows that
(5.2) lim KdA
t-0 2zr
2rr
Ir
KdA 2.
On the other hand, it is clearly the case that for W C U a compact region not
containing the double point
lim Iv KdA .fv
t--* O W V W
KdA
Setting Vt[] Vt B[0, ] where B[0, ] is the e-ball around the origin, we
deduce a special case of the theorem of Langevin (1
(5.3) lira lira
0 0 2r Jvt[l
1 l KdA 2.
This result establishes in principle the basic link between curvature and singularities.
We note that (5.3) has the following consequence due to Linda Ness2: Since,
by the discussion in section 4(b) (which in this case is quite obvious), the areas
(5.4) dA 0()
--
jvt[]
tend uniformly to zero, we deduce that there must be points p V where the
Gaussian curvature K(pt) -o. Moreover, because K _< 0 we may use (5.4) to
estimate the size of the region where K _< C < 0. It will be more convenient
for us to do this indirectly using the theory of currents.
These considerations are closely related to the HOcker paradox, which arose
in the very early days of algebraic geometry and brought into clear focus the
necessity for exercising caution in treating singularities. Suppose that C c IW
is an algebraic plane curve given in affine coordinates byf(x, y) 0 wherefis a
polynomial of degree d. The dual curve C* c IW* is defined to be the set of
tangent lines to C; i.e., the image under the Gauss mapping
C.
T: C--> Ip2..
It is again an algebraic curve of some degree d*, called the class of C, and the
dual of C* is again
To compute d* we recall that the degree of any algebraic curve is the number
of its intersections with a general line. By projective duality a line in IW* is
given by the pencil pl(p) of lines L through a fixed point p in IW. We may
choose our coordinates so that p [0, 0, 1] is the point at infinity along the yaxis
and such that the line at infinity is not tangent to C. The pencil P(p) then
consists of the vertical lines in the affine plane 2, and the class d* is just the