133

A method for computing generic Lê numbers
associated with non‐isolated hypersurface singulrities
By

Shinichi TAJIMA *Abstract

Lê cycles and Lê numbers introduced by D. Massey are considered in the context of symbolic
computation. A method for computing generic Lê numbers is proposed. Keys of the proposed
method are the use of parametric saturations in polynomial rings and of parametric local
cohomology systems.

§1.

Introduction

In 1991, D. Massey studied non‐isolated hypersurface singularities and introduced

the concept of Lê cycles and that of Lê numbers ([8], [9]). The Lê numbers are gener‐
alization of the Milnor number. D. Massey showed, among other things, in particularthat the alternating sum of Lê numbers is equal to the leduced Euler characteristic of

the Milnor fibre. He also gave in [8], [9], a method for computing Lê cycles and Lênumbers. However, as Lê numbers depend on the choice of coordinate systems used incomputation, they are not invariants of singularities. In contrast, generic Lê numbers

are complex analytic invariants of singularities (remark 9.1 in [10]). A problem comesfrom fact that no effective way for computing generic Lê numbers is known.

In a series of papers, by using the langage of derived category and the theory ofperverse sheaf and micro‐support, D. Massey has developed and generalised the theoryof Lê cycles and Lê numbers in tnore general context. Nowadays, Lê cycles and Lê

numbers are extensively studied by several authors ([1], [2], [4], [6]). Note in particular,
as T. Gaffney pointed out, that Lê cycles and generic Lê numbers are closely relatedwith holonomic D ‐modules associated with hypersurface singularities ([3], [10]) It istherefore desirable to establish an effective method for computing generic Lê numbers.
20ı0 Mathematics Subject Classification(s): Primary 32S05 , Secondary 13P10.68W30.
Key Words: polar variety, Lê cycle, parametric local cohomology system*

Supported by JSPS Grant‐in‐Aid No.
Univ. of Tsukuba, Japan

15KT0102,15K04891

134

SHINICHI TAJIMA

We propose in this paper an effective method for computing generic Lê numbers.
The main idea of our approach is the use of a familly of coordinate systems. Key tools
are parametric Gröbner systems [13], [19] and parametric local cohomology systems[15]. We show that these two tools allow us to compute generic Lê numbers withoutchoosing a generic coordinate system.
§2.

Polar variety and Lê cycle

In this section, we recall some basics on polar varieties and Lê cycles.
Let X be an open neighbourhood of the origin \mathcal{O} in \mathbb{C}^{n+1} Let h be a holomorphic
function defined on X , S the hypersurface S=\{x\in X|h(x)=0\} defined by h . Let\Sigma_{h} denote the singular set of S :\Sigma_{h}=\{x\in S|h(x)=\frac{\partial h}{\partial x_{0} (x)=\frac{\partial h}{\partial x_{1} (x)=. . =\frac{\partial h}{\partial x_{n} (x)=0\}.
Let s be the dimension at \mathcal{O} of the singular set \Sigma_{h}.

Now let us breafly recall a method given by D. Massey for computing L\hat{c} cycles andLê numbers. Suppose that a system of coordinates z= (z_{1}, z_{1}, \ldots, z_{n}) is given. AssuInethat it is generic enough.

Remark D. Massey introduced in [8] several notion of genericity. We refer thereader to [8], [9] for details.
For s<k\leq n , set

J_{h,z}^{(k)}= ( \frac{\partial h}{\partial z_{k} , \frac{\partial h}{\partial z_{k+1} \frac{\partial h}{\partial z_{n} ), I_{\Gam a_{hz} ^{(k)}=J_{h_{:}z ^{(k)}\subset \mathcal{O}_{X}
Z_{h,z}^{(k)}=V(J_{h,z}^{(k)}) , \Gamma_{h,z}^{(k)}=Z_{h,z}^{(k)}.
For

k=s , set

J_{hz}^{(s)}=(\frac{\partialh}{\partialz_{s},I_{\Gam a_{hz}^{(s+1)} , I_{\Gamma_{hz} ^{(s)}=J_{h,z}^{(s)} : I_{\Sigma_{h}^{\infty} . (saturation)
Z_{h,z}^{(s)}=V(J_{h,z}^{(s)}) , \Gamma_{hz}^{(s.)}=V(I_{\Gamma_{hz}}^{(s)},)and

I_{h,z}^{(s)}=J_{h,z}(s):(I_{\Gamma_{hz}}^{(s)})^{\infty} \Lambda_{h,z}^{(s)}=V (I_{h,z}^{(s)})
For 0<k<s , set

J_{h,z}^{(k)}=( \frac{\partial h}{\partial z_{k} , I_{\Gamma_{hz} ^{(k+1)}.

I_{\Gamma_{hz} ^{(k)}=J_{h,z}^{(k)}:I_{\Sigma_{h^{\dot{} } ^{\infty}.

GENERIC

135

I_{\lrcorner}\hat{E} NUMHERS

Z_{h,z}^{(k)}=V(J_{h,z}^{(k)}) , \Gamma_{h,z}^{(k)}=V(I_{\Gamma_{hz}}^{(k)})
and

I_{h,z}^{(k)}=J_{h,z}(k):(I_{\Gamma_{hz}}^{(k)})^{\infty}, \Lambda_{h,z}^{(k)}= V(I_{h,z}^{(k)})
For.

k=0 , set

J_{h,z}^{(0)}=( \frac{\partial h}{\partial z_{0} .I_{\Gamma_{hz} ^{(1)}) , Z_{h,z}^{(0)}=V(J_{hz}^{(0)}),

and

I_{h,z}^{(0)}=J_{h.z}(0) , \Lambda_{h,z}^{(0)}=V(I_{h,z}^{(0)}).

\Gamma_{h.z}^{(k)} and \Lambda_{h,z}^{(k)} are called polar variety and Lê cycles (or Lê variety) respectively.
Under the genericity condition, we have

Proposition 2.1. ([8], [9])k)

(ii)
(iii)

\dim\Lambda_{h,z}^{(k)}=k\Gamma_{h_{:}z}^{(k+1)}=\bigcup_{i\leq k}\Lambda_{h,z}^{(i)}\Sigma_{h}=\bigcup_{k\leq s}\Lambda_{h,z}^{(k)}The intersection numbers at the origin \mathcal{O}\gamma_{h,z}^{(k)}=(V(z_{0}, z_{1}, \ldots, z_{k-1})\cdot\Gamma_{hz}^{(k)}) _{\mathcal{O} , \lambda_{h,z}^{(k)}=(V(z_{0}, z_{1}, \ldots, z_{k-1}) \cdot\Lambda_{h,z}^{(k)})_{\mathcal{O} ,are called polar multiplicity and Lê number respectively.
Note that if we define\zeta_{h,z}^{k)}=(V(z_{0}, z_{1\cdots:}z_{k-1})\cdot Z_{hz}^{(k)})_{\mathcal{O} }then we have

Proposition 2.2. ([10])\zeta_{h,z}^{(k)}=\gamma_{h,z}^{(k)}+\lambda_{h,z}^{(k)},(ii) \zeta_{h,z}^{(0)}=\lambda_{h,z}^{0)}(i)

1\leq k

The result above will be used in the next section for computing generic Lê numbers.

The following example is taken from a paper of A. Zaharia [20].
Example 2.3.

Let h (x_{1}, x_{2}, y_{1}, y_{2})=y_{1}^{2}(y_{1}+x_{1}^{3}+x_{2}^{2})+y_{2}^{2} and set S=\{x\in \mathbb{C}^{4}|h(z)=0\} , wherex=(x_{1}, x_{2}, y_{1}, y_{2}) . The singular locus \Sigma_{h,x} of the hypersurface S is\Sigma_{h.x}=\{ (x_{1}, x_{2}, 0,0)|x_{1}, x_{2}\in \mathbb{C}\}\cong \mathbb{C}^{2}\subset \mathbb{C}^{4}
The dimension s of \Sigma_{h,x} is equal to 2.

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SHINICHI TAJIXIA

J_{h,x}^{(3)}=( \frac{\partial h}{\partial y_{2} )=(y_{2}) , Z_{h,x}^{(3)}=V(J_ {h_{)}x}^{(3)}),

I_{\Gamma_{hx}}^{(3)}=(y_{2}) , \Gamma_{h,x}^{(3)}= {( ı :^{x_{2:}y_{1},0)}|x_{1}, x_{2}.y_{1}\in \mathbb{C}\}.
xSinceJ_{h,x}^{(2)}=( \frac{\partial h}{\partial y_{1} , y_{2})=(2(x_{1}^{3}+x_{2} ^{2})y_{1}+3y_{1}^{2}, y_{2}),we have

I_{\Gamma_{hx}}^{(2)}=J_{h,x}^{(2)}:I_{\Sigma}^{\infty}=(2(x_{1}^{3}+x_{2}^{2}) +3y_{1}, y_{2}) , I_{h,x}^{(2)}=(y_{1}, y_{2}),and\Gamma_{h_{)}x}^{(2)}=\{ (x_{1}, x_{2}, y_{1}, 0)|2(x_{1}^{3}+x_{2}^{2})+3y_{1} =0\},\Lambda_{h,x}^{(2)}=\{(x_{1} , x_{2}, 0,0)|x_{1} : x_{2}\in \mathbb{C}\}.

From(x_{1}, x_{2}, I_{\Gamma_{hx}}^{(2)})=(x_{1}, x_{2}, y_{1}, y_{2}) , (x_{1}, x_ {2}, I_{hx}^{(2)})=(x_{1}, x_{2}, y_{1}, y_{2}),thc polar mmltiplicity \gamma_{hx}^{(2)} and the Lê number \lambda_{hx}^{(2)} are\gamma_{h,x}^{(2)}=\dim_{\mathbb{C} (\mathcal{O}_{X}/(x_{1} , x_{2}, y_{1}, y_{2})=1
As\lambda_{h,x}^{(2)}=\dim_{\mathbb{C} (\mathcal{O}_{X}/(x_{1} , x_{2}, y_{1}, y_ {2})=1.

,

J_{h,x}^{(1)}= ( \frac{\partial h}{\partial x_{2} , I_{\Gamma_{hx}}^{(2)})=(x_{2}y_{1}^{2}, 2(x_{1}^{3}+x_{2}^{2})+3y_{1}, y_{2}) ,I_{\Sigma}^{\infty}=(2x_{1}^{3}+3y_{1:}x_{2}, y_{2}) ,I_{h.x}^{(1)}=J_{h,x}^{(1)} : (I_{\Gamma_{hx}}^{(1)})^{\infty}=(2x_{1}^{3}+2x_{2}^{2}+3y_{1}, y_{1}^{2}, y_{2}) .

(x_{1}, x_{2}, 2x_{1}^{3}+3y_{1} , y_{2})=(x_{1} , x_{2} , yı , y_{2}) :(x_{1}, 2x_{{\imath}}^{3}+x_{2}^{2}+3y_{1} , y_{1}^{2}, y_{2})=(x_{1}, 2x_{2} ^{2}+3y_{1}, y_{1}^{2} : y_{2}) ,I_{\Gamma_{hx} ^{(1)},=J_{h,x}^{(1)}:Fromwe have\gamma_{h,x}^{(1)}=\dim_{C}(\mathcal{O}_{X}/(x_{1}, x_{2}, y_{1}, y_{2})=1,\lambda_{h,x}^{(1)}=\dim_{C}(\mathcal{O}_{X}/(x_{1}, 2x_{2}^{2}+3y_{1}.y_{1} ^{2}, y_{2})=4.

\Gamma_{h_{\backslash }x}^{({\imath})}=\{ (x_{1}, 0, y_{1} , 0)|2x_{1}^{3}+ 3y_{1}=0\}
Finally,,\Lambda_{hx}^{(1)}=\{(x_{1} , x_{2}, 0,0)|x_{1}^{3}+x_{2}^{2}=0\}.

J_{h,x}^{(0)}= ( \frac{\partial h}{\partial x_{1} , x_{2},2x_{1}^{3}+3y_{1}, y_{2})=(x_{1}^{2} y_{1}^{2}, x_{2},2x_{1}^{3}+3y_{1}, y_{2}) and I_{h,x}^{(0)}=J_{h,x}^{(0)},we have\lambda_{hx}^{(0)}=8
by direct computation.

\lambda_{h,x}^{(2)}, \lambda_{h,x}^{(\imath}) : \lambda_{h,x}^{(0)} are 1, 4, 8. Note that since \zeta_{h,x}^{(2)}. \zeta_{h,x}^{(1)} and \zeta_{h,x}^{(0)} , are equal
to 2.5 and 8, it follows from (\gamma_{h,x}^{(2)}, \gamma_{h,x}^{(1)})=(1,1) that (\lambda_{h,x}^{(2)}, \lambda_{h,x}^{(1)}, \lambda_{h,x}^{(0)})=(1 , 4, 8 ) imme‐
diately. Note also, as a set we have \Lambda_{h,x}^{(1)}=\Gamma_{h,x}^{(2)}\cap\Sigma_{h}.
Lê numbers

137GENERIC LÊ NUMBERS

For a relation with holonomic D ‐modules associated with b ‐functions, we refer thereaders to [18].
§3.

algorithm

We give an outline of an algorithm for computing generic Lê numbers. The main ideaof the proposed method is the use of a family of linear change of coordinate systems.

Key tools utilized to realise the idea above are parametric Gröbner systems [13], [14]and parametric local cohomology systems [15], [17], [19].
x_{n} ) in \mathbb{C}^{n+1} , we set z=(z_{0}, z_{1}, \ldots, z_{n})For a given system of coordinates, x= ( x_{0}, xı
by
x_{0}=z_{0}+t_{0} , lzı +t_{0,2}z_{2}+\cdot\cdot\cdot+t_{0,n}z_{n}x_{1}=z_{1}+t_{1,2}z_{2}+t_{1,3}z_{3}+\cdots+t_{1,n}z_{n}

.

.

.

=\cdotsx_{n}=z_{n}where t_{i,j} are parameters.
Algorithm

inputh(x) : polynomial(\lambda^{(s)} , \lambda^{(s-1)}, \ldots, \lambda^{(1)}, \lambda^{(0)}) : generic Lê numbers

outputstepl. compute the radical of the Jacobi ideal

I_{\Sigma x}=\sqrt{(\frac{\partial h}{\partial x_{0} ,\frac{\partial h} {\partial x_{1} \frac{\partial h}{\partial x_{n} )}step2. compute the dimension s at \mathcal{O} of the singular set \Sigma.
step3. I_{\Sigma} : rewrite I_{\Sigma} . in terms of variablez.

step4. set

J_{h}^{(s+1)}= ( \frac{\partial h}{\partial z_{s+1} , \frac{\partial.h}{\partial z.+2}, \frac{ \partial h}{\partial z_{n} ) I_{\Gamma_{h} ^{(s+1)}=J_{h}^{(s+1)}

step5. fork=s to 1,

J_{h}^{(k)}= ( \frac{\partial h}{\partial z_{k} , I_{\Gamma_{h}^{(k+1)} ), I_{\Gamma_{h} ^{(k)}=J_{h}^{(k)} : I_{\Sigma}^{\infty} (saturation)\zeta^{k}=(V(z_{0}, z_{1}, \ldots.z_{k-1})\cdot Z_{h}^{(k)})_{\mathcal{O} , \gamma^{(k)}=(V(z_{0}, z_{1}, \ldots, z_{k-1})\cdot\Gamma_{h}^{(k)}) _{\mathcal{O} ,where

Z_{h}^{(k)}=V(J_{h}^{(k)}), \Gamma_{h}^{(k)}=V(I_{\Gamma_{h} ^{(k)}).

\lambda^{(k)}=\zeta^{(k)}-\gamma^{(k)}, |\Lambda^{(k)}|=\Gamma_{h}^{(k)} \cap\Sigma
step6, set J_{h}^{(0)}= ( \frac{\partial h}{\partial z_{o} , I_{\Gam a_{h} ^{(1)} ,compute\lambda^{(0)}=multiplicity_{\mathcal{O}}(J_{j}^{(0)}), |\Lambda^{(0)}|138

SHINICHI TAJIMA

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