2 H. B. LAUFER
and this result was shown by Shepherd-Barron [46, Corollary 6, p. 7] with m 4.
The canonical map Sm ® Sn -» Sm+„ is surjective for m ^ 2 and « 3 (Theorems
3.2 and 3.5). This includes the m = « = 3 case, as conjectured by Reid (see [46,
Corollary 5, p. 5]).
Now suppose that (F, p) is normal and Gorenstein. Let IT\ M - F be the
minimal resolution of K The Gorenstein condition implies that we may take K to
be supported on A, the exceptional set in M. Then K- K is defined, and depends
on choosing the minimal resolution. Then A: °V-* 7 has a very weak simultaneous
resolution (Definition 4.1) after a finite base change if and only if Kt-Kt is
constant (Theorem 5.7). For multiple singularities, sum the individual Kt-Kr
Some important aspects of the proof of Theorem 5.7 were done by Shepherd-Bar-
ron  under the hypothesis that K has no base points. Theorem 5.7 is proved as
follows. We first take T to be 1-dimensional and X to be algebraic. Then the Sm
fit together to form a coherent sheaf of ideals im on T. The K • K condition
implies that im is normally flat. A construction similar to blowing up Ta t im gives
the simultaneous RDP resolution. This resolves to a very weak simultaneous
resolution after a finite base change . The result for arbitrary reduced T then
follows from general considerations [10, 40, 28].
-Kt - Kt is upper semicontinuous (Theorem 5.2). Let ht = dim Hx(Mn 0). Then
Kt • Kt constant implies that ht is constant (Theorem 5.3).
If Vt has a singularity/?, such that (Vn pt) is homeomorphic to (F, /?), then X
has a weak simultaneous resolution (Theorem 6.4). No base change is required via
Wahl's result . The proof of Theorem 6.4 makes essential use of Neumann's
The author would like to thank Professors Hironaka, Lipman, Schlessinger,
Teissier and Wahl for useful and stimulating conversations.
II. Known preliminaries. Throughout this paper, dim denotes dimension as a
complex vector space.
Recall  how the Riemann-Roch theorem may be extended to the one-di-
mensional singular case. Let C be a compact one-dimensional (reduced) analytic
space all of whose singularities are plane curve singularities, i.e. C can be locally
embedded in two-dimensional manifolds. Let m\ X ~ C be a resolution, i.e. a
normalization, of C. Let Sc be the structure sheaf on C. Let Gx be the structure
sheaf on X. 77^(6^) is then the sheaf of germs of weakly holomorphic functions on
C. Let S '•= 7T^(GX)/6C. Then 2 is supported precisely at the singular points of
C. Let 8 : = 2 dim Sx, where the sum is over all singular points x of C. Let c be
the conductor [24, p. 116]. Then there is a natural inclusion
Then [24, Theorem 1.1, p. 115] also 8 = S d i m ^ , where
the sum is again over all singular points of C. Observe that the degree of the
divisor c is 28.
Let L be a line bundle over C. Let 0(L) be the sheaf of germs of sections of L.
Let Lx be the pull-back of L of X. Let &(LX) be the sheaf of germs of sections of