数理解析研究所講究録
1111 巻 1999 年 52-57

52

HYPERSURFACES HAVING HIGHER ORDER CONTACT WITHSINGULAR SPACESGUANGFENG JIANG AND ARON SIMIS
ABSTRACT. This note is to study collections of functions whose zero level hypersurfacesof $X$ . These
are tangent to the regular part of a singular space $X$ along a subspace
to $X$ .
relativeof
ideal
defining
the
ideal
of
primitive
higherthe9collections are called
We establish the connection between higher iterated primitive ideals and the (relative)
and give an effective algorithm to compute both, where
symbolic powers of the idealis the defining ideal of $X$ .
$Y$

$Y$

$9/\mathfrak{h}$$\mathfrak{h}$, defined by ideals
Let $X\supset Y$ be germs of analytic subspaces of$Y$ are reduced and pure dimensional.
$X$
The mainand, respectively. Assume that
object of this note is to study a higher order analog of the so-called primitive ideal of 9-Siersma (cf. [5], [6], [2]).
relative to $X$ introduced byideal of a relative to $X$ consists of all funcprimitivethetheory,In terms of singularity$X$
and relative to the logarithmic stratification of $X$ )tions with critical loci (restricted to$Y$ . Geometrically, this ideal collects all functions whose zero level hypersurfaces
containing. From geometric point
along
pass through $Y$ and are tangent tozerolevel hypersurfaces
whoseof
functionsthecollections
study
to
is
reassonableof view, it$X$$Y$. These collections
along
with
pass through and have higher order of contact
are called higher relative primitive ideals. There are actually two notions of higher primitive
ideals in the relative case due to the fact that higher (algebraic) differential operators ona singular space are not just the iterated composites of differential operators of order one.
Since this intrinsic difficulty leads to a few delicate points in the theory, we made a choice
to work with the iterated version.
, the higher primitive ideals coincide with the higher symbolic powers
In case of(phased out by one) of the ideal -this is a consequence of the well-known Nagata-Zariski
theorem on holomorphic functions. It therefore seems quite natural to ask if the present-bears likewise a close relation
notion of higher primitive ideals in the relative case. This paperin the residue class ringto the higher symbolic powers of the ideal
deals systematically with this question for higher order. We answer this question under areasonable condition: the critical locus of $X$ does not contain any irreducible components
of Y. We also show that the higher (iterated) primitve ideals are effectively computablein terms of syzygies of certain matrices. As a bonus, we answer under certain conditions
a question posed by Vasconcelos (cf. [9]) as to how actually compute (relative) symbolic
powers in an effective way.
In studying functions with non-isolated singularities on singular spaces, we found thatthe topoplogy of the Milnor fibre of a function depends also on how its singular locus $Y$ is$\mathfrak{h}\subset \mathrm{g}\subset O=$$(\mathbb{C}^{n}, 0)$$O_{\mathbb{C}^{n},0}$$\mathrm{J}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}-\mathrm{P}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{k}\mathrm{a}\mathrm{a}\mathrm{n}$$Y\cap X_{\mathrm{r}\mathrm{e}\mathrm{g}}$$X_{\mathrm{r}\mathrm{e}\mathrm{g}}=X\backslash X_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$$Y\cap X_{\mathrm{r}\mathrm{e}\mathrm{g}}$$X=\mathbb{C}^{n}$$\mathrm{g}$$\mathfrak{h}\subset \mathrm{g}$$\mathrm{g}/\mathfrak{h}$$O/\mathfrak{h}$.
; Secondary
1991 Mathematics Subject Classification. Primarysyebolic
ideal,
space,
primitive
Key words and phrases. hypersurface, higher order contact, singularpower.
The ffist author was supported by Japan Society for Promotion of Science: P98028.
The second author was Partially supported by CNPq, Brazil.
$13\mathrm{B}10,14\mathrm{H}20$$13\mathrm{N}10,13\mathrm{P}99,16\mathrm{S}32,32\mathrm{S}05$

53

GUANGFENG JIANG AND ARON SIMIS

embedded in $X$ . There arises a probleIn: how to distinguish the different)
of $Y$
in $X$ ? de include somc examples to show that the higher (iterated) primitive ideals can beused to tell how le sits in $X$ .
$\mathrm{e}\mathrm{m}\mathrm{f}$$\mathrm{e}\mathrm{d}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}$VECTOR FIELDS OF HIGHER ORDER CONTACTLet
be a commutative ring and let $k\subset O$ be a subring. The setof kmodule endomorphisms of is an $O$ -module in the usual way whereby scalar multiplication
is to be taken as the composite, withdenoting theendomorphism multiplication by .
Given an integer $r\geq 0$ , let Dffi )denote the O-submodule
consisting of the differential operators of orderof the -algebra . The traditionalreference for differential operators of algebras is [1] (cf. also [4]). Our main interest willstem from the following notion.
$O$$\mathrm{E}\mathrm{n}\mathrm{d}_{k}(O)$$O$$a\rho(a\in O_{7}\rho\in \mathrm{E}\mathrm{n}\mathrm{d}_{k}(O))$$\mu_{a}\circ\rho$$\mu_{a}$

$a$

$r$

$(O)=\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{l}_{k}^{\langle r)}(O)\subset \mathrm{E}\mathrm{n}\mathrm{d}_{k}(O)$Definition 1. Let$\mathfrak{h}\subset O$

$O$

$k$

$\leq r$be an ideal. The $rth$ differential idealizer of$\mathrm{I}\mathrm{I}_{\mathfrak{h}}^{(r)}:=\{\delta\in \mathrm{D}\mathrm{f}\mathrm{f}\mathrm{i}^{\{r)}(\mathcal{O})|\delta(\mathfrak{h})\subset \mathfrak{h}\}$$\mathfrak{h}$is.

If no confusion arises, we will refer to thedifferential idealizer simply as the idealizer(of order ) .
the structure of -module ofas above easily follows that theidealizer of order of is an -submodule of. We haveO.
Since, where Der $(r)(O)$ is the -module of -derivations of order$r\mathrm{t}\mathrm{h}$

$r$

$O$

$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$$r$$\mathfrak{h}$$\mathrm{E}\mathrm{n}\mathrm{d}_{k}(O)$$O$$\mathrm{D}\mathrm{i}\mathrm{f}l^{\{r)}(O)$$\mathrm{D}ffi^{r)}(O)=O\oplus \mathrm{D}\mathrm{e}\mathrm{r}^{(r)}(\mathcal{O})$

$r$

of

$\mathit{0}$, then$\mathrm{I}\mathrm{I}_{\mathfrak{h}}^{(r)}=O\oplus \mathrm{I}\overline{\mathrm{I}_{\mathfrak{h}}^{(r)}}$$\mathrm{D}\mathrm{f}\mathrm{f}\mathrm{i}^{\{r)}(O)=\mathrm{I}_{\mathfrak{h}}^{(0)}=$$k$$\mathit{0}$, where. The latter module is$\mathrm{I}_{\mathfrak{h}}^{\overline{(r})}=\{\delta\in \mathrm{D}\mathrm{e}\mathrm{r}^{(r)}(O)|\delta(\mathfrak{h})\subset \mathfrak{h}\}$the essential part of the differential idealizer. Throughout we set).
$r\geq0$
For a given, we consider yet another -module, namely, the -submodule
generated by the elements of . In other words, the elements of thisare the-linear combinations of composites of elements of for values $t\leq r$ .
$\mathrm{I}\mathrm{I}_{\mathfrak{h}}=\mathrm{I}\mathrm{I}_{\mathfrak{h}}^{(1)}$$O$$\mathcal{O}$$(\mathrm{I}\mathrm{I}_{\mathfrak{h}})^{r}\subset$$\mathrm{I}\mathrm{I}_{\mathfrak{h}}^{(r)}$$\mathrm{s}\mathrm{u}\mathrm{b}_{\mathrm{I}}\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}$$\mathrm{I}\mathrm{I}_{\mathfrak{h}}$$\mathcal{O}$$t$$\mathrm{I}\mathrm{I}_{\mathfrak{h}}$Remark 2. Most of the results in this paper are valid for any field of characteristic zero.
To keep it more geometric, we work in the case whereandis a polynomial ring
(resp. a power series ring). Then one can interpret Der $(O)=$as (algebraic) vector fields on the affine space(resp. germs of analytic vector), then the essential part of the differential idealizer of order 1 of is to be
fields atinterpreted as the module of vector fields tangent to the subvariety(resp. to thegerm (V, ) of the analytic subspace at ) defined by the ideal . In case is a hypersurface(a divisor), these vector fields were dubbed logarithmic vector fields along(cf. [7]). For
higher values of , (the essential part of) the differential idealizer yields a notion of vector
fields of higher order of contact along .
$k$$k=\mathbb{C}$$\mathbb{C}[z_{1}, \ldots, z_{n}]$$\mathbb{C}\{z_{1},$$\ldots,$$\mathrm{D}\mathrm{e}\mathrm{r}^{(1)}(\mathcal{O})$$\mathcal{O}$$z_{n}\}$$\mathbb{C}^{n}$$0\in \mathbb{C}^{n}$$\mathfrak{h}$$V\subset \mathbb{C}^{n}$

$0$

$0$

$\mathfrak{h}$

$V$

$V$

$r$

$V$

HIGHER PRIMITIVE IDEALSDefinition 3. (i) Let
be ideals, with
$rth$ primitive ideal of(relative to $X$ ) is$\mathfrak{h}\subset \mathrm{g}\subset O$$\mathfrak{h}$radical and let $r\geq 0$ be an integer. The$\mathrm{g}$$\oint_{\mathfrak{h}}^{(r)}\mathrm{g}:=\{f\in \mathrm{g}|\mathrm{I}\mathrm{I}_{\mathfrak{h}}^{(r)}(f)\subset \mathrm{g}\}$.

54

HYPERSURFACES HAVING HIGHER ORDER CONTACT WITH SINGULAR SPACES(ii) The $rt,h$ iterated $pr\cdot imitive$ ideal of 9 relative to $X$ is$\int_{\mathfrak{h}}^{(r)}\mathfrak{g}:=\{f\in \mathrm{g}|(\mathrm{I}\mathrm{I}_{\mathfrak{h}})^{r}(f)\subset \mathrm{g}\}$Clearly,$\oint_{\mathfrak{h}}^{(r)}\mathrm{g}\subset\int_{\mathfrak{h}}^{(r)}\mathrm{g}$and, trivially,$\oint_{\mathfrak{h}}^{(0)}\mathrm{g}=\int_{\mathfrak{h}}^{(0)}\mathrm{g}=\mathrm{g}$.

. If $r=1$ , we omit the superscript,. Note that for this value of the present definition coincides
with that introduced by Pellikaan (cf. [5], [6], also [2]). Fixing the ideal , the question as
and for all values ofthroughout for all ideals
to when one has equalities
be a regular ring.
is rather subtle. We suspect it may be very close to implying thet
operators.
differential
Nakai
on
of
conjecture
known
well
to
the
relatedalso
isThe question
as well as
theoreticalaspects,in
As it turns, the first notion is rather unwielding several
computational. The clear advantage of the second notion is that it is recursively defined(see Proposition 7).
We now establish the connections between the iterated primitive ideal and the symbolic
powers of a given ideal. Recall that for any commutative noetherian ring and any ideal$I\subset A$ , we have the notion of the sth symbolic powerof , where is a nonegative
relative to theis the intersection of the primary components ofinteger. Namely,$I^{(s)}=$$xa\in I^{s}$
,
with
}.
minimal primes of . Equivalently,{
setting$r$$\oint_{\mathfrak{h}}^{(1)}\mathfrak{g}=\int_{\mathfrak{h}}^{(1)}\mathrm{g}=\int_{\mathfrak{h}}g$$\mathfrak{h}$$\mathrm{g}\supset \mathfrak{h}$$\oint_{\mathfrak{h}}^{(r)}\mathrm{g}=\int_{\mathfrak{h}}^{(r)}\mathrm{g}$$\mathcal{O}/\mathfrak{h}$

$r$

$A$

$I^{(s)}$

$I$

$s$

$I^{(s)}$$I^{s}$$I$

Proposition 4. Let(i) For every $r\geq 0$(ii) If$\mathrm{g}$$a \in A|\exists x\in A\backslash \bigcup_{P\in{\rm Min}(A/I)}P$

defined by, the support of the -module$\mathrm{Y}\subset X\subset \mathbb{C}^{n}$$i,deals\mathrm{g}\supset \mathfrak{h}$, with $X$ reduced. Then:is contained in$\int_{\mathfrak{h}}^{(r)}\mathrm{g}/\int_{\mathfrak{h}}^{(7+1)}\mathrm{g}$$\mathit{0}$has no embedded associated primes then$Ass(O/ \mathrm{g})=Ass(O/\int_{\mathfrak{h}}^{(r)}\mathrm{g})$. $r\geq 0$ .

(iii) If$\mathrm{g}$has no embedded associated primes then, for every $r\geq 0,$

where
(iv)

denotes the inverse image in.
, where$\overline{\mathrm{g}^{(r+1)}}$$\{\int_{\mathfrak{h}}^{(r)}\mathfrak{g}\}_{r\geq 0}$of.

for every$\mathrm{g}^{(r+1)}\subset\overline{\mathrm{g}^{(r+1)}}\subset\int_{\mathfrak{h}}^{(r)}\mathrm{g}$$(\mathrm{g}/\mathfrak{h})^{(r+1)}\subset O/\mathfrak{h}$,.

$0=\mathfrak{h}/\mathfrak{h}$$( \int_{\mathfrak{h}}^{(r)}\mathrm{g})/\mathfrak{h}=\int_{0}^{(r)}(\mathrm{g}/\mathfrak{h})$(v) The sequence$\mathit{0}$$Y$

is a descending multiplicative filtration.

Proof. (i) For every $r\geq 0$ the stalks of the module at the points outside are trivial.
containing(ii) It follows from the definition that, for any two ideals. One may stick to the case where is a primary ideal. Assuming$Y$$\mathfrak{h},$$\mathrm{g}_{1},$$\mathrm{g}_{2})=/_{\mathfrak{h}}(r)91\cap\int_{\mathfrak{h}}^{(r)}\mathrm{g}_{2}$$\int_{\mathfrak{h}}^{(r)}(\mathrm{g}_{1}\cap$$\mathrm{g}_{2}$$\mathrm{g}$is P-primary.
, one can easily prove thatof the symbolic powers. One
thedefinition
fromtrivially(iii) The first inclusion follows
can use (i) and (ii) to prove the second inclusion by induction on .
(iv)This follows immediately from the definitions and ffom the recursive character ofboth sides of the required equality.
(v) This follows from the definition.
We will next state a result in terms of geometric conditions. A similar result holds fora polynomial ring over a feild of characteristic zero.
defined by a radical ideal (sheaf) ,Recall that for a reduced analytic subspace $X$ ofis theof . In fact,
is well defined by the Jacobian ideal
the critical locus
jacobianoftheminorstheby
idealgenerated
is
the
,
where
ideal
, a generator set of , where stands for the codimension of $X$ . Notematrix ofthat this is a Fitting ideal, hence it is independent of the choice of generators of .

$Ass(O/\mathfrak{g})=\{P\}$$\mathrm{g}/\int_{\mathfrak{h}}^{(r)}\mathrm{g}$$r$$\square$

$O$

$k$

$\mathbb{C}^{n}$$X_{sing}$$(\tilde{\mathrm{J}}(\mathfrak{h})+\mathfrak{h})/\mathfrak{h}$$\{f_{1}, \ldots, f_{m}\}$$\mathfrak{h}$$\tilde{\mathrm{J}}(\mathfrak{h})$$\mathfrak{h}$$\mathfrak{J}(\mathfrak{h})$$c\cross c$$\tilde{\mathrm{J}}(\mathfrak{h})\subset O$$\mathfrak{h}$$c$$\mathfrak{h}$55

GUANGFENG JIANG AND ARON SIMIS

Proposition 5. Let

defined by radical ideals
does not contain any irreducible components of thenbe reduced, subspaces of$\cdot$$1^{r}\subset X$

respectively.
for every $r\geq 0$ .

$If_{-}1_{6ing}^{\Gamma}$$\mathbb{C}^{n}$$Y$

Proof. The inclusion$\mathfrak{h}\subset \mathrm{g}\underline{\subset O,}$$\int_{1)}^{(7)}\mathrm{g}=\mathrm{g}^{(r+1)}$

has been noted in Proposition 4, (iii). To prove the$\overline{\mathrm{g}^{(r+1)}}\subset\int_{\mathfrak{h}}^{(r)}\mathrm{g}$reverse inclusion it is sufficient to prove, the latter isbydefinition.
It is easy to see that taking primitive ideals commutes with localization (cf. [8]). Similarly,
it is a known fact that taking symbolic powers commutes with localization. Therefore, ateach point $P\in X_{reg}$ , one)
for every $r\geq 0$ . Hence outside$\mathrm{h}\mathrm{a}\underline{\mathrm{s}((\int}_{\mathfrak{h}}^{(r)}\mathrm{g}$$X_{sing}$

3, we have$:=\tilde{\mathrm{J}}(\mathfrak{h})$$( \int_{\mathfrak{h}}^{(r)}\mathrm{g})/\mathfrak{h}=\mathrm{g}^{(\gamma+1)}/\mathfrak{h}$annihilates$\underline{\int_{\mathfrak{h}}^{(r)}\mathrm{g}}/\mathrm{g}^{(r+1)}$$( \int_{\mathfrak{h}}^{(r)}\mathrm{g})/\mathfrak{h}=\mathrm{g}^{(r+1)}/\mathfrak{h}$$(\mathrm{g}/\mathfrak{h})^{(r+1)}$$/\mathfrak{h})_{P}=(\mathrm{g}/\mathfrak{h})_{P}^{(r+1)}$

everywhere. By R\"uckert Nullstellensatz, a power ofon$\mathcal{O}$as both ideals contain$\mathfrak{h}$by definition. In otherwords, we get. However, under the present hypothesis on:andhave the same associated primes.
the condition that $X_{sing}$ does not contain any$\mathrm{g},\overline{\mathrm{g}^{(r+1)}}$$J^{\infty}$$\int_{\mathfrak{h}}^{(r)}\mathrm{g}\subset \mathrm{g}^{(r+1)}$$\mathrm{g}$$\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s},-$irreducible components of$Y$implies that$\mathrm{g}^{(r+1)}$:$J^{\infty}=\overline{\mathrm{g}^{(r+1)}}$.

$\square$

THE COMPUTATION OF THE PRIMITIVE IDEAL

In this section we state two results without proofs to show how to effectively compute
theiterated primitive ideal. The basic computational result refers to the first primitive
ideal. We make the convention of writing the elements of the $O$ -submodule
as column vectors with coordinates in .
$r\mathrm{t}\mathrm{h}$$\mathrm{I}\mathrm{I}_{\mathfrak{h}}\subset Der(O)=$$O$$\sum_{i=1}^{n}Odz_{i}$Proposition 6. Let
be ideals, with radical. Let $\mathrm{g}=(g)=(g_{1}, \ldots, g_{m})$ be
generators of , letdenote the jacobian matrix of and let
denote the matrix
whose columns are a set of generators of the $O$ -submodule. Then$\mathfrak{h}\subset \mathrm{g}\subset \mathcal{O}$$\mathrm{g}$$\mathfrak{h}$$\mathrm{O}-(g)$$\triangle(\mathfrak{h})$$g$$\mathrm{I}\mathrm{I}_{\mathfrak{h}}\subset Der(\mathcal{O})$$\int_{1)}\mathrm{g}=I_{1}(g\cdot\Psi)$

where denotes the lifling to
the ideal .
$\Psi$$O$,of the syzygy matrix of the transpose$of\ominus(g)\cdot\triangle(\mathfrak{h})$modulo$\mathrm{g}$Proposition 7. Let$\int_{\mathfrak{h}}^{(r-1)}\mathrm{g}\}$$\mathfrak{h}\subset \mathrm{g}\subset O$. And consequently

be ideals, withradical. Then$\mathfrak{h}$$\int_{\mathfrak{h}}^{(r)}\mathrm{g}=\int_{\mathfrak{h}}(\int_{\mathfrak{h}}^{(r-1)}\mathrm{g})$$\int_{\mathfrak{h}}^{(r)}\mathrm{g}=\{f\in \mathrm{g}|\mathrm{I}\mathrm{I}_{\mathfrak{h}}(f)\subset$.

Remark 8. By shifting emphasis, the result of Proposition 5 can be viewed as a means ofcomputing the relative symbolic power. It is quite effective by Proposition 7. This partially answers a question posed by Vasconcelos (cf. [9]) as to whether one
can compute symbolic powers of an ideal in a residue ring of a polynomial ring in a relatively fast effective way (that avoids, say, taking the unmixed part of the ordinary powers
by means of computing withmodules and resolutions).
$(\mathrm{g}/\mathfrak{h})^{(r+1)}\subset \mathcal{O}/\mathfrak{h}$$\mathrm{E}\mathrm{x}\mathrm{t}$APPLICATIONS
A couple of examples may indicate the applications of higher relative primitive ideals.
Let $X\supset Y$ be germs of reduced analytic subspaces in, defined by radical ideals
as in the earlier sections. The higher iterated primitive ideals roughly tell us$(\mathbb{C}^{n}, 0)$$\mathfrak{h}\subset \mathrm{g}\subset O$

56

HYPERSURFACES HAVING HIGHER ORDER CONTACT WITH SINGULAR SPACES

how $Y$ is embedcled in $X$ . The case when $X$ is smooth was considered by Pellikaan [5]. In
case $X$ is singular and $Y$ is a smooth curve, the (first) primitivc ideal was computed in [2].
It turns out that, for fixed , which is taken to be the first axis in the local coordinates, the (first) primitive ideal of stays the same for each $X$, i.e.,
ofwith an isolated complete intersection singularity and containing $Y$ (see also the examples
below). Also when $X$ is a two dimensional isolated singular hypersurface, the length of
was used as a relative invariant of
the torsion submodule of the conormal module of. It is quite conceivable that
$Y$ with respect to $X[3]$ . It was proved that
surface $X$ with an isolated
same
on
the
remains unchanged for different smooth curves
singularity. However, for higher iterated primitive ideals the story is quite different.
$Y$$\mathrm{g}=(y_{1}, \ldots, y_{n})$$\mathbb{C}^{n+1}$$\mathrm{g}$$\lambda$$T$$\mathrm{g}/\mathfrak{h}$$T=(/_{\mathfrak{h}}(1)\mathrm{g})/\mathrm{g}^{2}+\mathfrak{h}$$\lambda$

singularity and let
Example 1. Let $h_{k}=xy+(1/k+1)z^{k+1}$ be the well known
be line contained therein. Then$(y, z^{r+1})$
if $r\leq k$otherwise$A_{k}$$\mathrm{g}=(y, z)$$\int_{h_{k}}^{(r)}\mathrm{g}=\{$$y \int_{h_{k}}^{(r-k-1)}\mathfrak{g}+(h_{k})$Example 2. In the first example the line in consideration had $\lambda=1$ . If we take a line
is defined by $h_{l}=x^{l}y+z^{2}+yz$
on an $A_{2l-1}$ surface to be the -axis, thenwith(see [3]). In this case we have$\lambda=l$$X_{l}$$x$$\int_{h_{l}}^{(2k)}\mathrm{g}=y^{k}(y, z)+(h_{l})$surface
Example 3. Consider therespect to the line defined by the ideal$D_{5,2}$$D_{5}$$\int_{h_{1}}^{(1)}\mathrm{g}=(y, z^{2})$,,$\int_{h_{l}}^{(2k+1)}\mathrm{g}=y^{k}(y, z^{2})+(h_{l})$.

, with defining equation $h_{1}=x^{2}y+y^{4}+z^{2}$ with. Then$\mathrm{g}=(y, z)$$\int_{h_{1}}^{(2)}\mathrm{g}=y\mathrm{g}+(h_{1})$,$\int_{h_{1}}^{(r+1)}\mathrm{g}=(y^{r})+(h_{1}))$$(r>2)$ .

surface once more, only now with respect to a different line
Example 4. Consider a: $h_{2}=x^{2}y+xz^{2}+y^{2}$ . Here we obtain:of the surface isequationbeing the -axis. The$D_{5}$$D_{5,2}^{*}$$x$$\int_{h_{2}}^{(1)}\mathrm{g}=(y, z^{2})$,$\int_{h_{2}}^{(2)}\mathrm{g}=(yz, xy+z^{2})+(h_{2})$,$\int_{h_{2}}^{(3)}\mathrm{g}=(yz^{2}, xy+z^{2})+(h_{2})$,and, for $k\geq 1$ ,$\int_{h_{2}}^{(4k)}\mathrm{g}=(xy+z^{2})^{k}\mathrm{g}+(h_{2})$,$\int_{h_{2}}^{(4k+1)}\mathrm{g}=(xy+z^{2})^{k}\int_{h_{2}}^{(1)}\mathrm{g}+(h_{2})$$\int_{h_{2}}^{(4k+2)}\mathrm{g}=(xy+z^{2})^{k}\int_{h_{2}}^{(2)}\mathrm{g}+(h_{2})$,,$\int_{h_{2}}^{(4k+3)}\mathrm{g}=(xy+z^{2})^{k}\int_{h_{2}}^{(3)}\mathrm{g}+(h_{2})$

.

.
and
for bothWe note that the invariant
For higher primitive ideals one can introduce the corresponding torsion number$\lambda=2$$D_{5,2}$$\lambda_{r}.--\dim_{\mathbb{C}^{\frac{\int_{\mathfrak{h}}^{(r)}\mathrm{g}}{\mathrm{g}^{r+1}+\mathfrak{h}’}}}$$D_{5,2}^{*}$$r=0,1,$.
Clearly,and 5 forfor
A calculation shows that
distinguish between the two singularities.

$\ldots$.

$\lambda_{1}=\lambda$$\lambda_{2}=4$$D_{5,2}$$D_{5,2}^{*}$. Therefore, these higher invariants

57

GUANGFENG JIANG AND ARON SIMISREFERENCES[1] A. Grothendieck and J. Dieudonn\’e, \’Eilements de Ge’ome’trie Alg\’ebrique, IV Partie, Publ. Math$\mathrm{I}.\mathrm{H}.\mathrm{E}/\mathrm{S}$

32 (1967).
[2] G. Jiang, Functions with non-isolated singularities on singular spaces, Thesis, Universiteit Utrecht,
1997.
[3] G. Jiang, D. Siersma, Local embeddings of lines in singular hypersulfaces, Preprint 1050, Departmentof Mathematics at Utrecht University, 1998, to appear in Annales deFourier.
[4] Y. Nakai, High order derivations I, Osaka J. Math., 7 (1970), 1-27.
[5] R. Pellikaan, Hypersurface singularities and Iesolutions of Jacobi modules, Thesis, Rijkuniversiteit
Utrecht, 1985.
[6] R. Pellikaan, Finite determinacy of functions with non-isolated singularities, Proc. London Math. Soc.,(3)57 (1988), 357-382.
[7] K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo
Sect. 1A Math. 27 (1980), 265-291.
[8] P. Seibt, Differential filtrations and symbolic powers of regular primes, Math. Z. 166 (1979) 159-164.
[9] A. Simis, Effective computation of symbolic powers by jacobian matrices, Comm. in Algebra, 24 (1996),
3561-3565.
$L^{f}\dot{\iota}nstitut$

DEPARTMENT OF MATHEMATICS, JINZHOU NORMAL UNIVERSITY, JINZHOU, LIAONING, 121000, P. R.
CHINA

DEPARTAMENTO DE MATEM\’ATICA, CCEN, UNIVERSIDADE FEDERAL DE PERNAMBUCO, CIDADE UN1VERSIT\’A, BRAZIL, 50740-540 RECIFE,$\mathrm{R}\mathrm{I}\mathrm{A}$$\mathrm{P}\mathrm{E}$