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
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
3l :=
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
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