The generalization of Buchberger's algorithm presented in this paper has many applications. We just mention the computation of Milnor numbers, Tjurina numbers, local multiplicities, Buchsbaum-Rim and Polar multiplicities, first and second order deformations of isolated singularities, projections of families with affine fibres onto a local base space and, of course, all the usual ideal theoretic operations in a factorring Loc;SPMlt;K[x]/I such as intersection, ideal quotient and decision about ideal or radical membership. For further applications see [AMR]. Here we shall only explain how its implementation in SINGULAR helped to find a partial answer to Zariski's multiplicity question.
Let , , f(0) = 0, be a not constant convergent powerseries and mult the multiplicity of f (for the general definition of multiplicity see the end of this chapter). Let be a sufficiently small ball with centre 0 and , the hypersurface singularity defined by f. If is another powerseries and , then f and g (or X and Y) are called topologically equivalent if there exists a homeomorphism such that h(X) = Y. The topological type of f is its class with respect to topological equivalence.
Zariski asked in 1971 (cf. [Z]) whether two complex hypersurface singularities f and g with the same topological type have the same multiplicity.
Zariski's question (usually called Zariski's conjecture) is, in general, unsettled but the answer is known to be yes in the following special cases:
Recall that f is called semiquasihomogeneous if there exists an analytic change of coordinates and positive weights such that the sum of terms of smallest weighted degree has an isolated singularity.
The two series of examples
fta, b, c and
gta, b, c, d, e
in Chapter 3 were actually constructed to find a counter example to
Zariski's conjecture. The idea is as follows: let
For the above mentioned series ftabc and gtabcde the multiplicity is not constant. For t = 0 both series have non-degenerate Newton diagram and one can show that and and small and some restrictions on a, b, c, d, e). Since is several hundred or even several thousand there seemed to be a good chance for or to be constant. Using SINGULAR we were able to compute many of these Milnor numbers but neither nor were constant. (Actually, in most cases we obtained and .) None of the existing computer algebra systems were able to compute the standard basis of the ideal of partials of ft respectively gt for relevant cases. (Only the system Macaulay was able to do some cases with small a, b, c using Lazard's method but it needed hours or days, whereas SINGULAR needed seconds or minutes. In these cases the success of SINGULAR was mainly due to the HCtest.) The failure to find a counter example led to the following positive result which shows that the families ft and gt can never be a counter example.
Let ft(x) be a (1-parameter) holomorphic family of isolated hypersurface singularities, that is is an isolated critical point of ft for each t close to . The polar curve of such a family is the curve singularity in defined by the ideal .
Proof: We may assume that and then the polar curve(ft|H) is given by while polar curve is given by . Hence, the assumption is equivalent to .
We shall use the valuation test for -constant by Lê and Saito
constant for any holomorphic curve we have val , . Moreover, this is equivalent to ``'' replaced by ``>''. (val denotes the natural valuation with respect to s.)
Now let be any curve in . Then implies that val , .
Applying the valuation test to ft and to , the result follows.
Proof: Since ft has an isolated singularity we may add
terms of sufficiently high degree without changing the analytic type of ft.
If n = 3 we may replace gt by
gt(x1, x2) + x1N + x2N, N
sufficiently big, which has an isolated singularity and the
same multiplicity as
gt(x1, x2). Hence, in any case we may assume that gt has an
isolated singularity. Applying the preceding lemma to the hyperplane
But since Zariski's conjecture is true for plane curve singularities and for
deformations of semiquasihomogeneous singularities ([Gr], [OS]), mult(gt) is constant.
The Milnor number of an isolated singularity can be computed as the number of monomials in where I is the leading ideal of with respect to any ordering > such that xi < 1, . This follows from the following Corollary 5.4.
The reason why standard bases can be applied to compute certain invariants of algebraic varieties or singularities (given in terms of submodules ), is that for any monomial ordering on K[x]r we have:
In order to show a) we make the following construction:
be a standard basis of
identified with the point
For a weight vector
It is not difficult to see that there exists a weight vector (indeed almost all w will do) such that in w(gi) = c(gi) L(gi), , and, moreover, in w(I) = L(I).
We choose such a w and shall now construct the deformation from L(I) to
For we can write such that the weighted degree of each monomial of is . Let t be one extra variable and put
Let be the submodule generated by all , . On K[x,t]r we choose the product ordering with lex- on K[t]: if p > q or if p = q and .
With respect to this ordering we have and, moreover, . In particular, is a standard basis of .
Let , and K(t) the quotient field of K[t].
Proof: The statements regarding the special and the generic
fibres are easy. If
then the support of
surjective over Spec
K[t](t) and hence ([AK], V, Proposition 2.4)
it remains to show that t is a non-zero divisor of
1.11 we have
Proof: If < is a wellordering, the monomials not in L(I) are a basis of the free module (Theorem of Macaulay cf. [E]), hence the result. In general, it is easy to see that these monomial are linear independent modulo IR. (Use a standard basis of I and Corollary 1.11.)
Hence, if Rr/IR is finite dimensional, there are only finitely many monomials in . The proposition implies that is K[t](t)-free with these monomials as basis, hence they also generate Rr/IR.
Proof: I = Rr implies
L(I) = K[x]r, hence we may assume
Faithful flatness implies that
([AK], V, Proposition 2.10), hence the result.
Let us finish with a final remark about multiplicites in the local case:
Consider the local ring
R = K[x](x) with maximal ideal
M = Rr/IR a finitely generated R-module, where I is
given as a submodule of K[x]r by finitely many generators. Consider
It follows that