The Milnor fibre signature is not semi-continuous

Dmitry Kerner and András Némethi

Abstract. Consider the germ of an isolated surface singularity in (ℂ^{3}, 0). The corresponding Milnor fibre possesses the homology lattice (the integral middle homology with a natural symmetric intersection form).

An old question of A.Durfee (1978) asks: is the signature of this form non-increasing under degenerations? The present article answers negatively: We give examples of Newton non-degenerate families where the signature increases under degeneration.

1. Introduction

Let (X, 0) = (f^{−1}(0), 0)⊂(ℂ^{3}, 0) be the germ of a complex, locally analytic surface with isolated singularity at the origin. For a general introduction to the ‘Milnor package’ associated with (X, 0), see [Milnor68], [Dimca92], [AGLV93] or [Seade2006].

Let B⊂ℂ^{3} be a small ball centered at the origin. Consider a small deformation: X_{ϵ} = f^{−1}(ϵ) ∩ B, where ϵ ≠ 0 is sufficiently small with respect to the chosen radius of the ball. Then X_{ϵ} is smooth and the diffeomorphism type of the pair (B, X_{ϵ}) does not depend on ϵ ≠ 0. X_{ϵ} is called the Milnor fibre and it has the homotopy type of a bouquet ∨_{μ}S^{2} of middle dimensional spheres, where μ is the Milnor number, hence H_{2}(X_{ϵ}, ℤ) ≈ ℤ^{μ}. There is a natural symmetric intersection form on H_{2}(X_{ϵ}, ℤ). Its extension to H_{2}(X_{ϵ}, ℤ) ⊗ ℝ gives the triple (μ_{+}, μ_{0}, μ_{−}), the dimensions of maximal vector subspaces of H_{2}(X_{ϵ}, ℤ)⊗ℝ on which the form is positive definite, zero or negative definite.

Let (X_{t}, 0)⊂(ℂ^{3}, 0) × (ℂ_{t}^{1}, 0) be a holomorphic family of surface germs, such that (X_{t}, 0) has an isolated singularity for any t in the neighborhood of 0 ∈ ℂ_{t}^{1}. An old question of A.Durfee [Durfee, Conjecture 5.4], see also W.Neumann [N.T.R.82, Problems session, p.249], is about the behavior of signature sign = μ_{+}−μ_{−} of the corresponding intersection forms under the degeneration:

Is the signature non-increasing under the degeneration t → 0?

It seems that the question was not addressed previously. The common expectation was that the semi-continuity as above should hold, at least under some mild restrictions. And the counterexamples, if any, must be quite complicated.

Since any non–smooth singularity is realized as the degeneration of (x, y, z) ↦ x^{2} + y^{2} + z^{2} whose signature is −1, a positive answer to the above question would imply that the signature of any non–smooth isolated hypersurface surface singularity is negative, which is the (weak) Durfee Conjecture [Durfee, Conjecture 5.2].

The present article shows that any tentative proof of the Durfee Conjecture which is based on semi–continuity of the signature in this direct way, will fail.

Note also that the signature can be ‘almost’ determined from the spectrum, which has a rather strong semicontinuity behavior, cf. [Varchenko83], [Steenbrink85]. The contribution in the signature, which is not covered by the spectrum, is the equivariant signature corresponding to the eigenvalue one of the monodromy, which depends on the structure of the Jordan blocks associated with this eigenvalue. The point is that exactly these blocks are responsible for the non–semi–continuity of the signature; and they are the key targets of the present article as well.

It is quite a surprise (at least for us) that the semi-continuity is violated even for families of Newton-non-degenerate surfaces. (In fact, all our examples are degenerations of T_{p,q,r} singularities.)

D.K. was constantly supported by the Skirball postdoctoral fellowship of the Center of Advanced Studies in Mathematics (Mathematics Department of Ben Gurion University, Israel).

A.N. is partially supported by OTKA and TÉT Grants of the Hungarian Academy of Sciences.

Example 1.1. Consider the family of surfaces X_{t} = f_{t}^{−1}(0), where for any t ∈ ℂ

(1)  f_{t} = txyz + xyz(x + y + z) + x^{4}y + y^{4}z + z^{4}x.

Note that X_{t} is Newton-non-degenerate for any t. In fact, X_{t≠0} is equivalent with the T_{13,13,13}–singularity {xyz + x^{13} + y^{13} + z^{13} = 0}. Indeed, first by scaling of the coordinates, one gets that f_{t≠0} is contact equivalent to xyz + x^{4}y + y^{4}z + z^{4}x. Further, using the transformation

(2)  \begin{matrix}x → x + 4y^{2}z^{3} − 6yz^{6} + 4z^{9} + 29y^{11}z^{2}, \\ y → x + 4x^{3}z^{2} − 6x^{6}z + 4x^{9} + 29z^{11}x^{2}, \\ z → z + 4x^{2}y^{3} − 6xy^{6} + 4y^{9} + 29x^{11}y^{2} \end{matrix}

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..........................................................................................................................................  ŷ x̂  ẑ . ^{.....................................................................................................................................................................................................} .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...........................................................................................................................................................................................•  • • (0,−1,3)  (−1,3,0) (3,0,−1)^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}^{.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .} . ^{.....................................................................} . ^{......................................................................................................................}. ^{.....................................................................} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ^{...................................................................................................} The Newton diagram of \frac{f_{0}}{xyz}

and dividing by 1 + 82(x^{8}y^{2} + 82y^{8}z^{2} + 82z^{8}x^{2}) one gets xyz + x^{13} + y^{13} + z^{13} + ..., where the higher order terms consist of monomials of total degree at least 15.

The relevant numerical data for the family is:

(3)  \begin{aligned}& μ & μ_{+} & μ_{0} & μ_{−} & μ_{+} − μ_{−} \\ X_{0} & 45 & 5 & 3 & 37 & −32 \\ X_{t≠0} & 38 & 1 & 1 & 36 & −35 \end{aligned}

Once a counterexample is found a natural question is: how significantly the signature can grow in degenerations? Below we consider the families of Newton-non-degenerate surface singularities. We explain the idea behind example

1.1 and present additional interesting families. In particular, it appears that the signature can increase significantly. We give an example when the change of μ_{+} − μ_{−} is asymptotically \frac{μ(X_{0})}{12} (or \frac{μ(X_{t≠0})}{9}).

Remark 1.2. If one considers the family X_{t} as a deformation of the central fibre and allows the singular point to split into several ones then an immediate counterexample is obtained as the suspension of a Newton-non-degenerate curve singularity.

Let (C_{0}, 0)⊂(ℂ^{2}, 0) be a singularity of the topological type of x^{3} = y^{9}, i.e. three smooth branches, pairwise tangent with order 3. Consider the family C_{t} in which the initial singularity deforms to five singular points, of the types (A_{5}, A_{5}, A_{1}, A_{1}, A_{1}). An example of such a family is: f_{t}(x, y) = y(y − x^{3})(y + (x − t)^{3}). Here the A_{5} singularities are at the points (0, 0) and (t, 0), while the three nodes A_{1} are the intersection points of the branches {y − x^{3} = 0} and {y + (x − t)^{3} = 0}.

Let X_{t} = {z^{2} + f_{t}(x, y) = 0}⊂ℂ^{3} be the stabilization of such a family. The relevant data for the signature is:

(4)  \begin{aligned}& μ & μ_{+} & μ_{0} & μ_{−} & μ_{+} − μ_{−} \\ X_{0} & 16 & 2 & 0 & 14 & −12 \\ X_{t≠0} & 13 & 0 & 0 & 13 & −13 \end{aligned}

(where, for each fixed t, the invariants are summed up over all singular points of X_{t}). Hence sign(X_{0}) > sign(X_{t≠0}).

The original question [Durfee, Conjecture 5.6] and in [N.T.R.82] targets non–splitting families, hence from now on we consider only this case: Sing(X_{t}) = {0} for all t.

Acknowledgements. The results of this paper were obtained during the conference “Libgober 60”, Jaca, Spain (July 2009). We appreciate the hard labor of the organizers, excellent conditions and working atmosphere.

For numerical computations we used [GPS-Singular], in particular the library: ”Gauß-Manin connection” by M.Schulze. We appreciate the power of the program and the help of the Singular forum.

Finally, we thank the referee for the careful reading of the manuscript.

2. Preliminaries

2.1. The Milnor number. For isolated Newton-non-degenerate singularities various topological invariants are determined by the Newton diagram. It is defined as follows: first, one takes the Minkowski sum of the support of f and the first octant ℝ_{≥0}^{3}. Then the union of all the compact faces of the convex hull of this set is called the Newton diagram, and it is denoted by Γ_{f}.

The function f is Newton-non-degenerate or non-degenerate with respect to its diagram if for any face σ⊂Γ_{f} the restriction f|_{σ} is non-degenerate, i.e. the corresponding hypersurface has no singular points inside the maximal torus.

Let X = f^{−1}(0)⊂(ℂ^{3}, 0) be a Newton-non-degenerate germ with Newton diagram Γ_{f}. We can assume that Γ_{f} is convenient (by adding to f the monomials x^{N}, y^{N}, z^{N} for N ≫ 0).

Let ℝ_{≥0}^{2} ⊃ Γ_{xy} = Γ_{f} ∩ {z = 0}, similarly for Γ_{yz}, Γ_{xz}. Note that all these diagrams are convex and convenient. Let ℝ_{≥0}^{1} ⊃ Γ_{x} = Γ_{f} ∩ {z = 0 = y}, and similarly introduce Γ_{y}, Γ_{z}.

The Milnor number is determined by the volumes under Γ_{f} as follows [Kouchnirenko76]

(5)  μ = 3!V ol_{3} − 2!V ol_{2} + V ol_{1} − 1,

where V ol_{3} is the volume in ℝ_{≥0}^{3} under the Newton diagram Γ_{f}, V ol_{2} is the sum of volumes in ℝ_{≥0}^{2} under the diagrams Γ_{xy}, Γ_{yz}, Γ_{xz}, and V ol_{1} is the sum of volumes in ℝ_{≥0}^{1} under the diagrams Γ_{x}, Γ_{y}, Γ_{z}. All the volumes are normalized so that the volume of the unit cube (in any dimension) is 1.

2.2. (μ_{+}, μ_{0}, μ_{−}). Given a symmetric intersection form let M_{+}, M_{−}, M_{0} be the maximal subspaces of H_{2}(X_{ϵ}, ℝ) on which the symmetric form is positive definite, zero, negative definite. So μ_{−} = dim(M_{−}), and similarly for μ_{0}, μ_{+}.

Recall [Steenbrink76] that the spectrum of (X, 0) is a set of rational numbers in (−1, 2), with multiplicities. The spectrum is symmetric with respect to \frac{1}{2}. The values Sp ∩ (0, 1) correspond to M_{−}, those from (Sp ∩ (−1, 0)) ∪ (Sp ∩ (1, 2)) correspond to M_{+} and those from Sp ∩ {0, 1} correspond to M_{+} ∪ M_{0}. In this last case, the precise distribution into M_{+}, M_{0} is determined by the Jordan block structure of the monodromy (which is not codified in the spectrum). Indeed, the cardinality of Sp ∩ {0, 1} is the dimension of the generalized eigen-space of the monodromy corresponding to eigenvalue one. Let J_{n} denote the number of n × n Jordan blocks corresponding to the eigenvalue one. By the monodromy theorem n is always ≤ 2, see e.g. [DoornSteenbrink89] and the references therein. Then J_{1} contributes 1 to μ_{0} (and provides either the spectral number 0 or the spectral number 1; by the symmetry of the spectrum such blocks appear in ‘pair’, one of them having spectral number 0, the other 1), while J_{2} contributes 1 to μ_{0} and 1 to μ_{+} (and provides two spectral numbers, namely 0 and 1).

The cardinalities of the spectral numbers in different intervals (as above) are computed from the Newton diagram as follows [Saito88]:

• ♯(Sp ∩ (−1, 0)) equals the number of ℤ_{>0}^{3} points strictly below Γ_{f}.

• ♯(Sp ∩ {0}) equals the number of ℤ_{>0}^{3} points on Γ_{f}; hence:

• ♯(Sp ∩ (−1, 0]) equals the number of ℤ_{>0}^{3} points not above Γ_{f}. Hence, μ_{0} + μ_{+} is twice this number.

Regarding the Jordan block structure on the eigenvalue 1 part of the monodromy one has ([DoornSteenbrink89], p. 227):

• J_{2} is the number of points of ℤ_{>0}^{3} ∩ Γ_{f} which do not lie in the interior of a two-dimensional (compact) face of Γ_{f}.

Example 2.1. Let X = f^{−1}(0)⊂ℂ^{3} for f = xyz + x^{p} + y^{q} + z^{r}. There is only one point of ℤ_{>0}^{3} which does not lie above Γ_{f}, namely (1, 1, 1); hence μ_{+} + μ_{0} = 2. This point does not lie in the interior of any face, hence it contributes to J_{2} and μ_{0} = 1 = μ_{+}.

Applying Kouchnirenko’s formula (or in any other way) we get: μ = p + q + r − 1, μ_{−} = p + q + r − 3 and the signature μ_{+} − μ_{−} = 4 − p − q − r.

2.3. Numerical computations. The general procedures to compute the spectrum and the monodromy are realized in [GPS-Singular] (though they are often time consuming). For, example the code for the case above is:

LIB ”sing.lib”; %%%% Loading the basic library of singularities. LIB ”gmssing.lib”; %%%% Loading the library of Gauß-Manin connection. ring s = 0,(x,y,z),ds; %%%% Local ring in 3 variables is defined int p,q,r=4,5,6; %%%% Integer constants poly f = x ∗ y ∗ z + x^{p} + y^{q} + z^{r}; milnor(f); spectrum(f); monodromy(f);

3. The idea behind the examples

As we consider the families of Newton-non-degenerate singularities, we prescribe the degeneration by erasing some vertices. For simplicity, we will only describe a ‘primitive’ degeneration: just one vertex is erased. By assumption, the vertex does not belong to the interior of a face or an edge. Hence, it is enough to consider just the relevant part of the Newton diagram.

Let a ∈ ℤ_{>0}^{3} be the apex of a (strictly convex, solid) cone C. Erase the apex, get a new smaller body: C_{new} = Conv((C \ a) ∩ ℤ_{≥0}^{3}). Let ∂C, respectively ∂C_{new}, denote the boundary of C, respectively C_{new}. An integral point of ∂C_{new}\ ∂C is called new. We assume that there is no new point which lies in the interior of a 2–face of C_{new}. Consider the boundary of ∂C_{new} \ ∂C, it is a closed 1–dimensional piecewise linear loop. The interior points of its edges are called inner. Its vertices are called outer. Finally, let V denote the volume of C \ C_{new}.

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Using these notations and assumptions, the discussion of (2.2) implies the following:

Proposition 3.1. Let N_{inner}, N_{outer}, N_{new} be the number of inner, outer and new points. Then:

(6)  \begin{matrix}μ(X_{0}) − μ(X_{t≠0}) = 6V \\ μ_{0}(X_{0}) − μ_{0}(X_{t≠0}) = −1 − N_{inner} + N_{new} \\ μ_{+}(X_{0}) − μ_{+}(X_{t≠0}) = 1 + N_{inner} + N_{new} \\ μ_{−}(X_{0}) − μ_{−}(X_{t≠0}) = 6V − 2N_{new} \\ sign(X_{0}) − sign(X_{t≠0}) = −6V + N_{inner} + 3N_{new} + 1. \end{matrix}

Next, we estimate V. Take an arbitrary elementary integral triangulation of the 2-faces of ∂C_{new}\ ∂C, i.e. the vertices of each triangle belong to ℤ_{≥0}^{3} and there are no integral points in the interior of the triangle or of its edges. Then C\C_{new} is naturally subdivided into the pyramids with the triangles as bases and the apex a ∈ C. Each pyramid is elementary, i.e. its only integral points are its corners.

Lemma 3.2. Set δ := 6V − N_{outer} − N_{inner} − 2N_{new} + 2. Then δ ≥ 0. The equality occurs iff the volume of each of the pyramids above is \frac{1}{6}.

Proof: By the construction above we get that 6V ≥the number of pyramids=the number of triangles in ∂C_{new}\ ∂C. The later quantity is combinatorial, does not depend on a particular triangulation, and can be computed explicitly. ■

From the previous proposition we get sign(X_{0}) − sign(X_{t≠0}) = 3 − δ − N_{outer} + N_{new}. Therefore, unsing this lemma in order to get an interesting example (when this expression is ‘large’, or at least it is positive), N_{outer} should be small, N_{new} should be large and the pyramids above should have the minimal volume \frac{1}{6} (or as small as possible).

4. A degeneration with big increase of signature

A simple generalization of the example 1.1 is the following.

Proposition 4.1. The invariants for the family of surfaces (X_{t}, 0) = f_{t}^{−1}(0)⊂(ℂ^{3}, 0) with

(7)  f_{t} = txyz + x^{3k+3}y + x^{k+1}yz + z^{2}x + y^{2}z

are:

(8)  \begin{aligned}& μ & μ_{+} & μ_{0} & μ_{−} & μ_{+} − μ_{−} \\ X_{0} & 12k + 11 & 2k + 1 & 1 & 10k + 9 & −8k − 8 \\ X_{t≠0} & 9k + 11 & 1 & 1 & 9k + 9 & −9k − 8 \end{aligned}

So the signature grows by k, therefore asymptotically by \frac{μ(X_{0})}{12} (or \frac{μ(X_{t≠0})}{9}) when k ↦ ∞.

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................................................................................................................................................................................  ŷ x̂  ẑ . ^{...................................................................................................} . ^{..........................................................................................................................................................} . ........................................................................................................................................................................................................................................................... . ^{.......................................................................................................................................................} .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •  • • • (3k+2,0,−1) (0,−1,1)  (−1,1,0) The Newton diagram of \frac{f_{t}}{xyz}

Notice that X_{t≠0} is equivalent to the singularity T_{6k+5,3k+4,3}. This can be proved in many different ways, and, in fact, all these methods can be used in the similar statement of Example 1.1 as well. By the first method, one computes the resolution graph of the germ, e.g. by Oka’s algorithm [Oka87] (cf. also [BraunNémethi07]), and one gets a cyclic graph which characterizes the hypersurface T_{p,q,r} germs. Or, one can use the results [BraunNémethi07] on ‘equivalent Newton boundaries’ which identifies the triple (6k + 5, 3k + 4, 3) from the diagram. It is clear that f_{t≠0} is equivalent to xyz + x^{3k+3}y + z^{2}x + y^{2}z. Then consider the three 2–faces of the the diagram and notice that they intersect the three axes at (6k + 5, 0, 0), (0, 3k + 4, 0) and (0, 0, 3) respectively. Then use [BraunNémethi07, §3]. Finally, one can get the result by change of variables too.

Proof: For the computation of μ_{0}, μ_{+} we consider the polynomial \frac{f}{xyz}, then we look for the point of ℤ_{≥0}^{2} lying under/not above Γ_{\frac{f}{xyz}}. One sees that any such point is of the form (x, 0, 0). In addition no integral points lie in the interior of the faces or edges of Γ_{f} (or Γ_{\frac{f}{xyz}}). Hence:

• t ≠ 0. There is only one point: (0, 0, 0) ∈ Γ_{\frac{f}{xyz}} it creates J_{2}, so contributes 1 to μ_{0} and 1 to μ_{+}.

• t = 0. The points (j, 0, 0) for 0 ≤ j < k lie under Γ_{\frac{f}{xyz}} they contribute 2k to μ_{+}. The point (k, 0, 0) ∈ Γ_{\frac{f}{xyz}} contributes 1 to μ_{0} and 1 to μ_{+}.

The Milnor number can be computed by Kouchnirenko formula or in any other way, then one obtains μ_{−} too. ■

References

[AGLV93] V.I. Arnol’d, V.V. Goryunov, O.V. Lyashko, V.A. Vasil’ev, Singularity theory.I. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems. VI, Encyclopaedia Math. Sci., 6, Springer, Berlin, 1993]. Springer-Verlag, Berlin, 1998.

[BraunNémethi07] G. Braun and A. Némethi, Invariants of Newton non-degenerate surface singularities. Compos. Math. 143 (2007), no.

4, 1003–1036.

[Dimca92] A. Dimca, Singularities and topology of hypersurfaces. Universitext. Springer-Verlag, New York, 1992. [DoornSteenbrink89] M.G.M. van Doorn and J.H.M. Steenbrink, A supplement to the monodromy theorem. Abh. Math. Sem. Univ.

Hamburg 59 (1989), 225–233.

[Durfee] A. Durfee, The signature of smoothings of complex surface singularities. Math. Annalen 232 (1978), 85–98. [GPS-Singular] G.M. Greuel, G. Pfister, H. Schönemann, Singular 3.0.0. A Computer Algebra System for Polynomial Computations.

ZCA, University of Kaiserslautern (2005). http://www.singular.uni-kl.de.

[Milnor68] J. Milnor, Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61 Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1968.

[N.T.R.82] Nœuds, tresses et singularités. [Knots, braids and singularities] Proceedings of the seminar held in Plans-sur-Bex, March 27–April 2, 1982. Edited by C. Weber. Monographies de L’Enseignement Mathématique, 31. Université de Genève, L’Enseignement Mathématique, Geneva, 1983; (Available elletronically on http://www.math.bgu.ac.il/∽kernerdm/).

[Kouchnirenko76] A.G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor. Invent. Math. 32 (1976), no. 1, 1–31.

[Oka87] M. Oka, On the resolution of the hypersurface singularities. In Complex analytic singularities, Adv. Stud. Pure Math., 8, 405–436, North-Holland, Amsterdam, 1987.

[Saito88] M. Saito, Exponents and Newton polyhedra of isolated hypersurface singularities. Math. Ann. 281 (1988), no. 3, 411–417. [Seade2006] J. Seade, On the Topology of Isolated Singularities in Analytic Spaces. Progress in Mathematics 241, Birkhäuser 2006. [Steenbrink76] J.H.M. Steenbrink, Mixed Hodge structure on the vanishing cohomology. Real and complex singularities (Proc. Ninth Nordic

Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 525–563. Sijthoff and Noordhoff, Alphen aan den Rijn, 1977. [Steenbrink85] J.H.M. Steenbrink, Semicontinuity of the singularity spectrum. Invent. Math. 79 (1985), no. 3, 557–565. [Varchenko83] A.N.Varchenko, Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective

hypersurface. (Russian) Dokl. Akad. Nauk SSSR 270 (1983), no. 6, 1294–1297.

Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva 84105, Israel.

   E-mail address: kernerdm@math.bgu.ac.il

Rényi Institute of Mathematics, Budapest, Reáltanoda u. 13–15, 1053, Hungary

   E-mail address: nemethi@renyi.hu