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CURVATURE AND COMPLEX SINGULARITIES 507
Pliicker defect A is then an algebraic surface in the Grassmannian G(2, 4).
Recall that G(2, 4) has dimension four, and that there are two families of Schubert
cycles in the middle dimension which may be described as follows"
for a hyperplane H, E(H) is the set of
2-planes T such that T C H (cf. (4.10));
for a line L, E,(L) is the set of
2-planes T such that L C T.
The first intersection number #(A, H) describes the limit of the number 8 of
critical values in a pencil of sections
C,t St (h + H);
as such it has to do with the number of vanishing cycles in the pencil ]Cx,tlx and
therefore with the Milnor number of So (cf. the proof of (5.15)). The formula
__ Consideration of the other Schubert cycle EI,I(L) leads to the following geometric
interpretation: Under a generic projection (4 3, S goes to a surface
S’ having a finite number 5’ of isolated singularities, and for generic L this
number is # (A, EI,(L)). In a manner similar to (5.24) we infer from (5.14) that
(5.24) lim lim Cte I C2(’St)
0 0 JSt[]
is a consequence of (5.14) and is of the same character as (5.17).
(5.25) lim lim
’-0 t--*0
c--e I,t, {c(Sa,)- c(a,)} ’
Based on the following analogy it seems possible that some variant of 5’ will
have topological meaning: For a smooth algebraic surface W C IW the Chern
numbers c and c2 may both be calculated from the degree, so that e.g., c c2
is determined by c2. But for a non-degenerate smooth surface W C I these
two numbers are independent in the sense that neither one determines the other.
Thus it seems reasonable that, at least for those surfaces So c ([;4 which are
limits of smooth surfaces, the independent numbers cz and c c can be localized
to yield two distinct invariants.
(ii) Suppose now that {Ct} is a family of curves in an open set in 1I; 3 tending to
a limit curve Co having an isolated singularity at the origin. The Gaussian images
yt(C) c G(1, 3) IW give a family of analytic curves in the projective
plane with
lim y,(C,) A + y0(C0)
t--*0
where the algebraic curve A is the Plticker defect. Recalling that Ct-e,K dA is the