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 [46] 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

t

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 [8]. 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 [52]. The proof of Theorem 6.4 makes essential use of Neumann's

theorem [38].

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 [24] 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

7T^(6X(-C))

C

0

C

.

Let

3l :=

&C/TT*(6X(-C)).

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