THE EMBEDDED RESOLUTION OF f(x, y) + z^{2} : (ℂ^{3}, 0) → (ℂ, 0)

CHUNSHENG BAN, LEE J. MCEWAN, AND ANDRÁS NÉMETHI

1. Introduction

1.1. The goal of the present paper is the presentation of an “embedded resolution” of {f(x, y) + z^{2} = 0}, 0) ⊂ (ℂ^{3}, 0) using the method of Jung. In the first part of the introduction, we present the terminology and the strategy of the paper.

Let (Y, 0) be the germ of an analytic space. We say that a proper, surjective analytic map ϕ : Ỹ → U is a resolution of (Y, 0) (where U is a small representative of (Y, 0)), if Ỹ is smooth, ϕ^{−1}(U − Sing Y) is dense in Ỹ, and ϕ^{−1}(U − Sing Y) → U − Sing Y is an isomorphism.

More generally, if (Y, 0) is a local divisor in (ℂ^{n}, 0), we say that ϕ : X̃ → U is an embedded resolution of the pair (Y, 0) ⊂ (ℂ^{n}, 0) (where again U is a small representative of (ℂ^{n}, 0)), if X̃ is smooth, ϕ is an isomorphism above U − Sing Y, and ϕ^{−1}(Y) is a normal crossing divisor in X̃.

Jung’s strategy ([4], see also [7], [8]) gives a recipe how one can reduce the construction of the resolution of the n-dimensional space (Y, 0) to the case of the construction of an embedded resolution of (the “lower dimensional case”) (Δ, 0) ⊂ (ℂ^{n−1}, 0) and the resolution of the so-called quasi-ordinary singularities of dimension n. (For quasi-ordinary singularities, see [7], [8]). Indeed, consider a projection p : (Y, 0) → (ℂ^{n−1}, 0) with finite fibers, whose reduced discriminant locus is (Δ, 0). Then resolve the pair (Δ, 0) ⊂ (ℂ^{n−1}, 0) by ϕ : Z → U ⊂ ℂ^{n−1} and pull back p to p′ : Y′ → Z. Then Z is smooth and the branch locus of p′ is a normal crossing divisor, hence Y′ has only quasi-ordinary singularities. Resolving these singularities we obtain Ỹ.

For example, consider the hypersurface singularity (Y, 0) = ({f(x, y) + z^{2} = 0}, 0), where f is an isolated plane curve singularity. Then p : (Y, 0) → (ℂ^{2}, 0), given by (x, y, z) ↦ (x, y) is a double covering with branch locus ({f = 0}, 0) ⊂ (ℂ^{2}, 0). Using the above notations, if ϕ : Z → U ⊂ ℂ^{2} is an embedded resolution of this plane curve singularity, then Y′ normalized has only Hirzebruch-Jung type singularities of type {z^{2} = x^{m}y^{n}}^{norm}.

If we want to obtain an embedded resolution by a similar strategy, then we face two problems. To explain this, start with (Y, 0) ⊂ (ℂ^{n}, 0) and a projection p : (ℂ^{n}, 0) → (ℂ^{n−1}, 0) such that p|Y is finite. Let (Δ, 0) be the discriminant locus of p|Y. Then take an embedded resolution ϕ : Z → U^{n−1} ⊂ ℂ^{n−1} of the pair (Δ, 0) ⊂ (ℂ^{n−1}, 0) and construct by pull-back the diagram

X ⟶^{ϕ′} U^{n} ⊂ ℂ^{n}
p′↓  ↓^{p}
Z ⟶^{ϕ} U^{n−1} ⊂ ℂ^{n−1}

Then still we have to find the embedded resolution of the pair (ϕ′)^{−1}(Y) ⊂ X. Although (ϕ′)^{−1}(Y) has only quasi-ordinary singularities, in general we know very little about their embedded resolution.

Moreover, even if we solve this problem (i.e. we find an embedded resolution X̃ → X of these quasi-ordinary singularities), the final result of the above construction has a small beauty defect: the constructed birational map X̃ → U^{n} is not an isomorphism above the complement of Sing Y, but only above a smaller set, the complement of p^{−1}(Sing Δ).

For example, if (Y, 0) = ({f(x, y) + z^{2} = 0}, 0) ⊂ (ℂ^{3}, 0), and p(x, y, z) = (x, y), and we solve all the technical problems in the above program, then we obtain a map ϕ̃ : X̃ → U^{3} ⊂ ℂ^{3} so that ϕ̃^{−1}(Y) is a normal crossing divisor, but ϕ̃ is an isomorphism only above ℂ^{3} − {x = y = 0}, not above ℂ^{3} − {0}.

Even if this birational modification does not cover exactly the above definition of the embedded resolution, in almost all the applications it plays the role of an embedded resolution: all the invariants which can be read from an embedded resolution can be read from this modification as well.

The goal of the present paper is exactly the presentation of this birational modification (“embedded resolution”) of ({f(x, y) + z^{2} = 0}, 0) ⊂ (ℂ^{3}, 0).

We wish to emphasize: our goal is not to prove the existence of an embedded resolution (this is well–known, even in more general cases, see e.g. [2, 12]), but to give an explicit description of a possible modification together with all the topological data of the exceptional divisors, their intersection and embeddings.

Finally, we mention, that the (non-embedded) resolution of the normal surface singularity {f(x, y) + z^{n} = 0} is well-understood; see [5, 6, 9] for the shape of the dual resolution graph, [1] for the case n = 2, and [10, 11] for other particular cases.

1.2. Preliminary Remarks. Let g : (ℂ^{n}, 0) → (ℂ, 0) be the germ of an analytic function. Sometimes, an embedded resolution ϕ̃: X̃ → U of the pair ({g = 0}, 0) ⊂ (ℂ^{n}, 0) (or a birational modification as above) is called the “resolution of g”. E denotes the exceptional divisor, and E = E_{1} ∪ ⋯ ∪ E_{s} the decomposition of E into its irreducible components.

In our situation when g = f(x, y)+z^{2}, since ϕ̃ is an isomorphism above ℂ^{3}−{x = y = 0}, the compact irreducible exceptional divisors are situated above the origin (i.e. ϕ̃(E_{i}) = 0), while the non-compact irreducible exceptional divisors projects via ϕ̃ onto the disc {x = y = 0}.

When we want to codify the modification ϕ̃, we have to decide what kind of information we would like to extract from it. In general ϕ̃ : X̃ → U (or even E) carries a lot of analytic information which is impossible to codify in any topological, numerical or combinatorial object. But if we want to study the pair (g^{−1}(0), 0) ⊂ (ℂ^{n}, 0) from a topological point of view, it is enough to record only the topological/numerical/combinatorial invariants of ϕ̃. If we want to codify additional analytic information as well, then we face the problem of analytic (or algebraic) classification of varieties and vector bundles (which is basically an unsolved problem). Therefore, in general, we have to find the right compromise which is still satisfactory for our final goals.

Take, for example the case n = 2 and an arbitrary embedded resolution ϕ̃. All the topology is completely codified in the dual resolution graph of E ⊂ X̃ with the decorations {E_{i} · E_{i}}_{i} (the self-intersections in X̃) and the vanishing orders {m_{i}}_{i} of g ∘ ϕ̃ along E_{i}’s. Indeed, from this decorated graph the homeomorphism type of (X̃, E), or of (g^{−1}(0), 0) ⊂ (ℂ^{n}, 0), can be completely recovered by plumbing. If we want to recover the analytic type of the reduced exceptional divisor E, then the combinatorics of the graph cannot identify the position of the points ⋃_{j≠i} E_{j} ∩ E_{i} on E_{i}. Therefore, in this second level, if we wish to recover the isomorphic type of E from our codification, we need also to keep in this codification the pairs (E_{i}, ⋃_{j≠i} E_{j} ∩ E_{i})_{i}. On the other hand, at the third level, if one wants to codify all the analytic information about (X̃, E), then that is a rather difficult problem: the underlying topology, in general, carries many analytic structures which are parametrized by a rather complicated moduli space.

In higher dimensions, the problem is more complicated, and in general it is not clear at all what the good, convenient levels and codifications of the resolution are.

In the situation discussed in this article, when g(x, y, z) = f(x, y)+z^{2} : (ℂ^{3}, 0) → (ℂ, 0), it turns out that the codification of the modification of g constructed above is closely related to the codification of the embedded resolution of f. We will define the analog of the resolution graph of f for g, which will keep all the topological information about the pair (E ⊂ X̃) up to a homeomorphism. This graph will not identify E modulo an analytic isomorphism, but the ambiguity will be very similar to the plane curve singularity case. (In other words, g carries the same amount of analytic information as f.)

Actually, our modification ϕ̃ = ϕ_{g} of g will be constructed from a fixed embedded resolution ϕ = ϕ_{f} of f. If we denote the irreducible exceptional divisors of ϕ_{f} by {A_{i}}_{i}, then any compact irreducible exceptional divisor E_{k} of ϕ_{g} will be a (possibly non-minimal) ruled surface with natural projection E_{k} → A_{i} = ℙ^{1}. All the special fibers will be situated over the intersection points {A_{j} ∩ A_{i}}_{j≠i} ⊂ A_{i}. Similarly, the non-compact irreducible exceptional divisors will be non-minimal disc bundles over some curves A_{i}. If from the resolution of ϕ_{f} we retain the information of the position of the intersection points {A_{j} ∩ A_{i}}_{j≠i} ⊂ A_{i}, then the analytic type of the exceptional divisors E_{k} will be completely determined. If we use only the dual resolution graph of f, then in the analytic type of E_{k} we will have the ambiguity of the position of the special fibers. This ambiguity will disappear in the computation of any kind of numerical invariant, and in the identification of different elements (like K_{E_{k}} or N_{X̃|E_{k}}) in Pic (E_{k}) for any compact E_{k}.

2. Review of Ruled Surfaces over ℙ^{1}

For a general reference for ruled surfaces, we recommend [3].

2.1. Any ruled surface over a smooth curve is obtained as ℙ(E) of a locally free sheaf (vector bundle) E of rank 2. Actually, ℙ(E) = ℙ(E ⊗ L) for any line bundle L. But, over ℙ^{1}, any E can be written as E = O(a)⊕O(b), hence any ruled surface over ℙ^{1} has the “normal form” X_{e} = ℙ(O ⊗ O(−e)) for some integer e ≥ 0. The surface X_{e} can be obtained by gluing two copies of ℂ × ℙ^{1} (with coordinates (x, [u_{0} : u_{1}]) and (y, [w_{0} : w_{1}]) respectively) along ℂ^{∗} × ℙ^{1} by the identifications y = 1/x, and [w_{0} : w_{1}] = [u_{0} : x^{e}u_{1}]. One has a natural projection π : X_{e} → ℙ^{1} (in coordinates (x, [u_{0} : u_{1}]) ↦ x, (y, [w_{0} : w_{1}]) ↦ y). Here ℙ^{1} = ℂ ⨆_{ℂ^{∗}} ℂ, where the x-chart ℂ corresponds to {[α : β] | α ≠ 0} with x = β/α.

An automorphism ϕ : X_{e} → X_{e} with π ∘ ϕ = π is called a π-automorphism; their collection form the group G(X_{e}, π).

The projection π : X_{e} → ℙ^{1} has two natural sections with images C_{0} and C_{1}. C_{0} is given by {u_{1} = w_{1} = 0} and has self intersection C_{0}^{2} = −e; C_{1} is given by {u_{0} = w_{0} = 0} and has self intersection C_{1}^{2} = e. Obviously C_{0} ∩ C_{1} = ∅.

2.2. Facts. a) ([3], V. 2.3) Pic X_{e} = ℤ ⊕ π^{∗}Pic ℙ^{1} = ℤ ⊕ ℤ is generated by C_{0} and an arbitrary fixed fiber f of π. They satisfy C_{0} · f = 1, f^{2} = 0.

b) ([3], V 2.8) The invertible sheaf O_{X_{e}}(1) (provided by the Proj-construction of ℙ(E)) satisfies O_{X_{e}}(1) = O_{X_{e}}(C_{0}).

2.3. If im τ is the image of a section τ : ℙ^{1} → X_{e}, then im τ ≡ C_{0} + nf in Pic X_{e} for some n. The condition im τ ∩ C_{0} = ∅ implies (C_{0} + nf) · C_{0} = 0, hence n = e. In particular (im τ)^{2} = (C_{0} + ef)^{2} = e.

2.4. If e = 0, then X_{0} can be represented as π : X_{0} = ℙ^{1} × ℙ^{1} → ℙ^{1}, where π is the first projection and C_{0} and C_{1} are two fibers of the projection on the second factor. Obviously, the system (π : X_{0} → ℙ^{1}; C_{0}, C_{1}) is uniquely determined modulo the action of G(X_{0}, π) (i.e. it does not depend on the choice of C_{0} and C_{1}).

2.5. Assume that e > 0. In the sequel we will prove that the last sentence of (2.4) is valid in this case as well. First notice that the section C_{0} is uniquely determined by the condition C_{0}^{2} = −e (cf. [3], V 2.11.3). Therefore, Fact 2.2 implies that any ϕ ∈ G(X_{e}, π) keeps C_{0}, O_{X_{e}}(1), Pic X_{e} invariant.

By [3], V 2.6, there is a one-to-one correspondence between sections τ : ℙ^{1} → X_{e} of π and surjections O ⊕ O(−e) → L, where L ∈ Pic ℙ^{1}, given by L = τ^{∗}O_{X_{e}}(1). Under this correspondence, C_{0} corresponds to the second projection pr_{2} : O ⊕ O(−e) → O(−e) and C_{1} to the first projection pr_{1} : O ⊕ O(−e) → O.

Notice that it is possible to construct sections τ : ℙ^{1} → X_{e} with image C′_{1} := τ(ℙ^{1}), not identical to C_{1}, but satisfying the same numerical properties as C_{1}: C′_{1} ∩ C_{0} = ∅ and C′_{1}^{2} = e. Nevertheless, one has:

2.6. Proposition. Fix e > 0 and consider a ruled surface π : X_{e} → ℙ^{1} and C_{0}, C_{1} ⊂ X_{e} as above. Then take an arbitrary section τ with im τ = C′_{1} satisfying C′_{1} ∩ C_{0} = ∅ (hence also C′_{1} = C_{1}^{2} = e, cf. 2.3). Then there exists ϕ ∈ G(X_{e}, π) such that ϕ(C_{1}) = C′_{1}. In particular, the system (π : X_{e} → ℙ^{1}; C_{0}, C_{1}) is uniquely determined (up to isomorphism) by the integer e and the conditions C_{0}^{2} = −e, C_{0} ∩ C_{1} = ∅ (and from the fact that C_{0} and C_{1} are images of sections of π). C_{1} automatically satisfies C_{1}^{2} = e. Proof. Let g : O ⊕ O(−e) → L be the surjection corresponding to the section τ : ℙ^{1} → X_{e} with im τ = C′_{1}. Then by [3], V 2.9, deg L = C_{0} · C′_{1}, hence L = O_{ℙ^{1}}. Now, consider the map g ⊕ pr_{2} : O ⊕ O(−e) → O ⊕ O(−e). Since C_{0} ∩ C′_{1} = ∅, this is an isomorphism, which induces ϕ.

2.7. Example. (cf. [3] II 8.24.) Assume that two smooth surfaces E_{1} and E_{0} in the smooth three-fold X intersect each other transversally. Additionally, assume that E_{1} ∩ E_{0} = C is isomorphic to ℙ^{1} and N_{C|E_{1}} = O(a) and N_{C|E_{0}} = O(b) with a ≤ b. Let X̃ be obtained by blowing up X along C, and let E be the exceptional divisor of this blowing up π. Then π induces a projection π : E → ℙ^{1} making E a ruled surface. In fact E ≈ X_{b−a}.

ℙ^{1} C_{1}^{m} C_{0}^{m} −e P_{0}  P_{k} ⋯  ⋯  ⋯ P′_{j_{1}} m{′ \atop j_{1}} = 1  P′_{j_{2}} m{′ \atop j_{2}} = 2l^{j_{2}}  P′_{j_{3}} m{′ \atop j_{3}} = 2l^{j_{3}} + 1 ≥ 3 0 (1)  0 (1)  0 (1)  −1 (1) −2 (1) −2 (1) −1 (1) −2 (1)  −1 (1) −2 (1) −2 (1) −3 (1) −1 (2) −2 (1) π^{m}
Figure 1. The surface X_{e}^{m}

Let C_{i} be the intersection of E with the strict transform of E_{i} (i = 0, 1). Then C_{0} and C_{1} are exactly the irreducible curves on X_{b−a} determined by (2.6). In particular N_{C_{1}|E} = O(b − a) and N_{C_{0}|E} = O(a − b).

2.8. Non-minimal ruled surfaces. Now, we will fix a ruled surface π : X_{e} → ℙ^{1} and a set of distinct points {P_{1}, …, P_{k}; P′_{1}, …, P′_{l}} on ℙ^{1}. Let Q′_{j} be the unique point on C_{1} with π(Q′_{j}) = P′_{j}.

In the sequel we will modify X_{e} by some blow ups using the following recipe. Above the points {P_{j}}_{j} we will not modify the fibers, we will only mark them. On the other hand, the fibers π^{−1}(P′_{j}) will (eventually) be modified. For this, we fix some integers m′_{j} > 0 with 2e = ∑_{j=1}^{l} m′_{j}. This also means that if e = 0, then l = 0 and {P′_{1}, …, P′_{l}} = ∅, hence there is no modification. If e > 0, then we fix local coordinates (x_{j}, y_{j}) in a small neighborhood U_{j} of Q′_{j} with {x_{j} = 0} = C_{1} ∩ U_{j},

and we consider a local curve D = {x_{j}^{2} = y_{j}^{m′_{j}}} in U_{j}. Then we will modify U_{j} by a minimal sequence of blow ups so that the total transform of D ∪ C_{1} form a normal crossing divisor. Here we distinguish three cases. If m′_{j} = 1, then D = {x_{j}^{2} = y_{j}} intersects C_{1} transversally, hence no blow up is needed. If m′_{j} is odd and m′_{j} ≥ 3, then (m′_{j} + 3)/2 blow ups are needed. Finally, if m′_{j} is even, then one needs m′_{j}/2 blow ups. We do this for any 1 ≤ j ≤ l. Notice that the sequence of blow ups (in particular, the analytic type of the new surface) is independent of the choices of the local coordinates (x_{j}, y_{j}) in U_{j}, depending only on the integers m′_{j} (and the position of the points {P′_{j}}_{j} in ℙ^{1}). The new surface will be denoted by X_{e}^{m}. The surface X_{e}^{m} and the projection π^{m} : X_{e}^{m} → ℙ^{1} (where π^{m} = π∘the sequence of blow ups) looks as in Figure 1.

Here C_{i}^{m} denotes the strict transform of C_{i} (i = 0, 1). The strict transform of π^{−1}(P′_{j}) ⊂ X_{e} is the unique irreducible component of (π^{m})^{−1}(P′_{j}) which intersects C_{0}^{m}. The other components of (π^{m})^{−1}(P′_{j}) are the new exceptional divisors. The non-positive integer near each irreducible curve denotes the self-intersection of the curve in X_{e}^{m}. The positive integer in parenthesis is the multiplicity of the corresponding irreducible component in the divisor π^{∗}(P′_{j}).

It is clear that

2.9.

\{ \begin{matrix}(C_{0}^{m})^{2} & = & −e \\ (C_{1}^{m})^{2} & = & e − ∑_{m′_{j} even} m′_{j}/2 − ∑_{m′_{j} odd >1}(m′_{j} + 1)/2 \end{matrix}

Moreover, by [3], V 3.2, one has:

2.10. Pic (X_{e}^{m}) = ℤ^{r}, where r = 2 + ∑_{m′_{j} even} m′_{j}/2 + ∑_{m′_{j}odd >1}(m′_{j} + 3)/2

and it is generated by C_{0}^{m}, a generic fiber f, and the new irreducible exceptional curves {C′_{j}}. The intersection matrix can be easily read from Figure 1 using (C_{0}^{m})^{2} = −e, f^{2} = 0, C_{0}^{m} · f = 1, and C_{0}^{m} · C′_{j} = f · C′_{j} = 0 for any irreducible exceptional curve C′_{j}.

Notice that the isomorphism type of X_{e}^{m} is completely determined by the integer e, {m′_{j}}_{j=1}^{l} and the position of the points {P′_{j}}. On the other hand, the homeomorphism type of the system (X_{e}^{m}, C_{0}^{m}, C_{1}^{m}, {(π^{m})^{−1}(P_{j})}_{j}, {(π^{m})^{−1}(P′_{j})}_{j}) is determined completely by e and {m′_{j}}, and does not depend on the choice of the points {P_{j}}_{j} and {P′_{j}}_{j}.

2.11. Any compact irreducible exceptional divisor of the birational modification which will be constructed later, associated with g = f(x, y) + z^{2}, will be isomorphic to some X_{e}^{m}. The irreducible components of C_{0}^{m} ∪ C_{1}^{m} ∪ (∪_{j}(π^{m})^{−1}(P_{j})) ∪ (∪_{j}(π^{m})^{−1}(P′_{j})) provide the intersection curves with other irreducible exceptional divisors.

On some of the irreducible exceptional divisors X_{e}^{m} we have to put one more curve: the intersection of X_{e}^{m} with the strict transform of {g = 0}. This discussion is postponed until (3.5).

2.12. The non-compact irreducible exceptional divisors of g will be (non-minimal) disc bundles over ℙ^{1}. Notice that for any e ∈ ℤ, there is a unique disc bundle π : B_{e} → ℙ^{1} such that the zero section C satisfies C^{2} = e. Similarly as in the case of X_{e}, we can fix some data which codifies a sequence of blow ups with centers above the zero section of π. In this way, we obtain the (non-minimal) modified disc bundle B_{e}^{m} with natural projection B_{e}^{m} → ℙ^{1}. These are the candidates for the non-compact irreducible exceptional divisors of our birational modification.

3. The Embedded Resolution of g = f(x, y) + z^{2}

3.1. First we fix some notations regarding a convenient embedded resolution graph of the isolated plane curve singularity f : (ℂ^{2}, 0) → (ℂ, 0).

Let ϕ_{f} : Z → U^{2} be an embedded resolution of the pair (f^{−1}(0), 0) ⊂ (ℂ^{2}, 0), where U^{2} is a small representative of (ℂ^{2}, 0). We denote the irreducible exceptional divisors by {A_{1}, …, A_{s}}, the strict transforms of the irreducible components of {f = 0} by {St_{1}, …, St_{s′}}, and the collection of all these irreducible components by {D_{1}, …, D_{s+s′}}. It is obvious that each A_{i} is rational. Topologically, the pair ϕ_{f}^{−1}({f = 0}) ⊂ Z (or ({f = 0}, 0) ⊂ (ℂ^{2}, 0)) is codified by the following data:

a) the intersection matrix (A_{i} · A_{j})_{i,j};

b) the intersections A_{i} · St_{j};

c) the multiplicities m_{i}(f) of f ∘ ϕ_{f} along D_{i} (actually a, b ⇒ c). In the next construction, it is convenient to assume that there is no pair D_{i} and D_{j} with D_{i} · D_{j} ≠ 0 such that both m_{i}(f) and m_{j}(f) are odd.

This situation can always be realized: indeed if D_{i}∩D_{j} ≠ ∅ and m_{i}(f)·m_{j}(f) ≡ 1 mod 2, then we blow up the intersection point D_{i} ∩ D_{j}. The multiplicity of the new exceptional divisor will be the even number m_{i}(f) + m_{j}(f). Notice that there is a unique minimal resolution ϕ_{f} with this property.

We will use the following notations as well. For a generic point P on A_{i}, there are local coordinates (u, v) in a small neighborhood U of P such that f ∘ϕ_{f}|U = u^{m_{i}(f)}. Similarly, if P = D_{i} ∩ D_{j}, then in some local coordinates in a small neighborhood U of P one has f ∘ ϕ_{f}|U = u^{m_{i}(f)}v^{m_{j}(f)}.

3.2. In the construction of the embedded resolution graph of g(x, y, z) = f(x, y) + z^{2}, we will use Jung’s strategy. Consider the following diagram (already mentioned above):

X̃ ⟶^{r} X ⟶^{ϕ′} U^{3} ⊃ g^{−1}(0)
↓^{p′}  ↓^{p}
Z ⟶^{ϕ_{f}} U^{2} ⊃ f^{−1}(0)

We have:

• U^{3} is a small representative (polydisc) of (ℂ^{3}, 0) and p : U^{3} → U^{2} is induced by the projection (x, y, z) ↦ (x, y).

• ϕ_{f} : Z → U^{2} is an embedded resolution of (f^{−1}(0), 0) ⊂ (ℂ^{2}, 0) as described in (3.1).

• p′ : X → Z is the pull-back of p : U^{3} → U^{2} via ϕ_{f}. Notice that X is smooth. Let T_{g} := (ϕ′)^{−1}(g^{−1}(0)) be the total transform of g^{−1}(0) in X. Fix a generic point A_{i} and small coordinate neighborhood U of P as in (3.1). Then (p′)^{−1}(U) admits local coordinates (u, v, z), where p′(u, v, z) = (u, v), and T_{g} ⊂ (p′)^{−1}(U) has equation u^{m_{i}(f)} +z^{2} = 0. Similarly, if P = D_{i} ∩D_{j}, then T_{g} is given by u^{m_{i}(f)}v^{m_{j}(f)} + z^{2} = 0.

• r is an embedded resolution of T_{g} ⊂ X (cf. 3.4). The composed map ϕ′∘r is denoted by ϕ_{g}. By construction ϕ_{g} : X̃ → U^{3} is an “embedded resolution” of (g^{−1}(0), 0) ⊂ (ℂ^{3}, 0), which is an isomorphism above U^{3} − {x = y = 0}, cf. Introduction.

Now we will describe the exceptional divisor E of ϕ_{g}.

3.3. The exceptional divisor E is a union E_{nc} ∪ E_{c} where E_{nc} (respectively E_{c}) is the union of non-compact (respectively, compact) irreducible exceptional components. E_{nc} is created in two steps: first we create the exceptional divisors of ϕ′, then we modify them by some blow ups. Indeed, the resolution ϕ_{f} gives rise to the exceptional curve A = ϕ_{f}^{−1}(0). This lifted, gives rise to the exceptional surfaces A × D = (ϕ′)^{−1}(D) ⊂ X, where D is the disc {x = y = 0} ⊂ U^{3}. Each irreducible component of A × D has the form A_{j} × D, i.e. it is a disc bundle with trivial self-intersection of the zero section. We denote A_{j} × D by E(A_{j}).

The multiplicity of the function g (or of z) along each E(A_{j}) is zero.

If A′_{j} = (p′|T_{g})^{−1}(A_{j}) ⊂ T_{g}, then N_{A′_{j}|X} = O ⊕ O(A_{j}^{2}).

3.4. Now we describe r : X̃ → X in more detail. First, we fix an ordering of the irreducible exceptional divisors {A_{j}}_{1≤j≤s} of ϕ_{f}. By convention, if m_{i}(f) is even and m_{j}(f) is odd, then i < j. (Sometimes we say that A_{i} is older than A_{j} if i < j.) Such an ordering always exists and in general it is not unique. Different orderings provide different modifications.

Now, fix an irreducible component A_{i}. Fix a generic point P on A_{i}, let U be a small neighborhood of P as in (3.1). Then T_{g} ∩ U = {(u, v, z) : u^{m_{i}(f)} + z^{2} = 0}, where {u = 0} = E(A_{i}) ∩ U.

The transversal plane curve singularity has type A_{m_{i}(f)−1}; corresponding to its minimal embedded resolution, we blow up the corresponding rational curves above A_{i}. First we resolve this transversal singularity completely above A_{1}, then we continue with A_{2}, and so on. If m_{i}(f) is even, we need m_{i}(f)/2 blow ups; if m_{i}(f) = 1 then we need no modification, if m_{i}(f) is odd and > 1, then we need (m_{i}(f) + 3)/2 blow ups.

More precisely, assume that we finished this procedure for A_{1}, …, A_{i−1}, and we want to continue with the curve A_{i}.

If m_{i}(f) is even, m_{i}(f) = 2l_{i}, then the graph of the minimal embedded resolution of ({u^{2l_{i}} + z^{2} = 0}, 0) ⊂ (ℂ, 0) is:

[figure: ⋯; (2); (4); (2l_{i} − 2); (2l_{i}); (1); (1); −2; −2; −2; −1]

where (n) denotes the multiplicity of u^{m_{i}(f)}+z^{2} along the corresponding irreducible exceptional divisor. Corresponding to this, we make l_{i} blow ups along the corresponding rational curves (as axis) above A_{i}. Let’s see what types of ruled surface we will obtain.

If A_{j} is older than A_{i}, i.e. j < i, then m_{j}(f) is automatically even, hence E(A_{i}) is modified ∑_{j<i,A_{i}A_{j}≠0} m_{j}(f)/2 times in different infinitesimally close points. After these modifications E(A_{i}) becomes E(A_{i})′. The strict transform of A′_{i} in E(A_{i})′ is denoted by A′′_{i}. Then A′′_{i} has normal bundle O(− ∑_{j<i} m_{j}(f)/2) ⊕ O(A_{i}^{2}). For simplicity, we write A_{i}^{2} = e_{i} and − ∑_{j<i} m_{j}(f)/2 = x_{i}. Then we start to resolve the tranversal singularity A_{m_{i}(f)−1} above A_{i}. After the sequence of blow ups along the rational curves above A_{i}, we obtain a tower of ruled surfaces above A_{i}, as it is shown in Figure 2, (cf. also with 2.7). In this diagram the schematic picture

[figure: e_{i}; x_{i} − l_{i}e_{i}; l_{i}e_{i} − x_{i}; x_{i} − (l_{i} − 1)e_{i}; (l_{i} − 1)e_{i} − x_{i}; x_{i} − 2e_{i}; 2e_{i} − x_{i}; x_{i} − e_{i}; e_{i} − x_{i}; x_{i}; X_{|l_{i}e_{i}−x_{i}|}; X_{|(l_{i}−1)e_{i}−x_{i}|}; X_{|2e_{i}−x_{i}|}; X_{|e_{i}−x_{i}|}; E^{m}(A_{i}); −1; −2; −2; −2; (2l_{i}); (2l_{i} − 2); (4); (2); ⋮; ⋮; ⋮; ⋮; ⋮; ⋮]

Figure 2. Tower of ruled surfaces, the case m_{i}(f) = 2l_{i}

n −n X_{|n|}

denotes the ruled surface X_{|n|}, the horizontal lines denote the two distinguished curves C_{0} and C_{1} ⊂ X_{|n|} with self-intersections ± n. In Figure 2 an adjacency shows that X_{|n|} and X_{|m|} intersect each other along their distinguished curve codified by the common horizontal line. The arrows on the graph of {u^{2l_{i}} +z^{2} = 0} correspond to the strict transforms; their contribution will be discussed in (3.5). Obviously, the integers (n) denote the vanishing orders of g along the corresponding components X_{|n|}.

Since the divisor (f) of f on Z satisfies (f) · A_{i} = 0, one has

e_{i}m_{i}(f) + ∑_{j<i,A_{j}A_{i}≠0} m_{j}(f) + ∑_{j>i,A_{j}A_{i}≠0} m_{j}(f) = 0.

Therefore, x_{i} − e_{i}l_{i} = \frac{1}{2} ∑_{j>i,A_{j}A_{i}≠0} m_{j}(f) ≥ 0.

Therefore, the surface at the bottom of the tower is X_{x_{i}−e_{i}l_{i}}. The collection {m_{j}(f)}_{j>i,A_{j}A_{i}≠0} is sometimes denoted by {m_{j}′(f)}_{j}. The bottom horizontal line is the distinguished curve C_{1} of X_{x_{i}−e_{i}l_{i}} (since it has positive self-intersection). All

[figure: e_{i}; x_{i} − (l_{i} + 1)e_{i}; (l_{i} + 1)e_{i} − x_{i}; 2x_{i} − (2l_{i} + 1)e_{i}; (2l_{i} + 1)e_{i} − 2x_{i}; x_{i} − l_{i}e_{i}; l_{i}e_{i} − x_{i}; x_{i} − (l_{i} − 1)e_{i}; (l_{i} − 1)e_{i} − x_{i}; x_{i} − e_{i}; e_{i} − x_{i}; x_{i}; X_{|(l_{i}+1)e_{i}−x_{i}|}; X_{|(2l_{i}+1)e_{i}−2x_{i}|}; X_{|l_{i}e_{i}−x_{i}|}; X_{|e_{i}−x_{i}|}; E^{m}(A_{i}); −2; −1; −3; −2; −2; −2; (2l_{i} + 1); (4l_{i} + 2); (2l_{i}); (2l_{i} − 2); (4); (2); ⋮; ⋮; ⋮; ⋮; ⋮; ⋮]

Figure 3. Tower of ruled surfaces, the case m_{i}(f) = 2l_{i} + 1

the other surfaces will be unchanged by the latter modifications, but this X_{x_{i}−e_{i}l_{i}} will be changed by some blowing ups corresponding to the modifications of the younger neighbors A_{j} (j > i) of A_{i}.

If m_{i}(f) = 2l_{i} + 1 is odd, then the graph of the minimal resolution of {u^{2l_{i}+1} + z^{2} = 0} is

[figure: (1); ⋯; (2); (4); (2l_{i} − 2); (2l_{i}); (4l_{i} + 2); (2l_{i} + 1); −2; −2; −2; −3; −1; −2]

Since all the neighbors of A_{i} are older than A_{i} (since all of them have even multiplicity), N_{A′_{i}|X} = O(x_{i}) ⊕ O(e_{i}) where x_{i} = − ∑_{i≠j,A_{i}A_{j}≠0} m_{j}(f)/2 and e_{i} = A_{i}^{2} as above. Then the tower of ruled surfaces is as in Figure 3.

D̃ −1 −2 −2 −2 −1 0 0 0 0 0 0 0 −1 −2 −2 −2 (2t_{j}) (2t_{j} − 2) (4) (2) (0) X_{x_{j}−t_{j}e_{j}}^{m} X_{|e_{j}−x_{j}|} E^{m}(A_{j})  ⋮  (2l_{i}) (2l_{i} − 2) (4) (2) (0) E^{m}(A_{i})  X_{x_{i}−l_{i}e_{i}}^{m} X_{|e_{i}−x_{i}|}^{m} ⋮
Figure 4. Gluing, the case m_{i}(f) = 2l_{i}, m_{j}(f) = 2t_{j} and j < i

The relation (f) · A_{i} = 0 now reads as

m_{i}(f)e_{i} + ∑_{A_{j}A_{i}≠0,i≠j} m_{j}(f) = 0

i.e. (2l_{i} + 1)e_{i} − 2x_{i} = 0. Therefore, the second surface from the bottom (which will support the strict transform corresponding to the arrow) is X_{0} = ℙ^{1} × ℙ^{1}.

Now we will analyze how we have to glue all these towers of surfaces. This discussion will clarify also how we have to modify X_{x_{i}−e_{i}l_{i}} (the case m_{i}(f) even) corresponding to the blowing ups of the younger neighbors.

If P is an intersection point A_{i} ∩ A_{j}, then T_{g} ⊂ (p′)^{−1}(U) has local equation u^{m_{i}(f)}v^{m_{j}(f)} + z^{2} = 0. We distinguish two cases. In the first case m_{i}(f) = 2l_{i}, m_{j}(f) = 2t_{j}, and we ssume that the u-axis is older than the v-axis (i.e. j < i). Then above U, the exceptional divisor E has the form shown as in Figure 4. In the picture, the D̃ is a disc over D = {x = y = 0}. The vertical segments of contact

D̃ −2 −1 −3 −2 −2 −1 0 0 0 0 0 0 0 0 −1 −2 −2 −2 (2t_{j}) (2t_{j} − 2) (4) (2) (0) X_{x_{j}−t_{j}e_{j}}^{m} X_{|e_{j}−x_{j}|} E^{m}(A_{j})  ⋮  (2l_{i} + 1) (4l_{i} + 2) (2l_{i}) (4) (2) (0) E^{m}(A_{i})  X_{|e_{i}−x_{i}|}^{m} ⋮
Figure 5. Gluing, the case m_{i}(f) = 2l_{i} + 1, m_{j}(f) = 2t_{j}

between projective surfaces denote rational curves. The integers a|b denote the self-intersection numbers of this curve in the two surfaces correspondingly.

If the local equation is u^{m_{i}(f)}v^{m_{j}(f)}+z^{2} = 0, where m_{i}(f) = 2l_{i}+1, m_{j}(f) = 2t_{j}, then automatically j < i, and E above a neighborhood of P looks as in Figure 5.

Now, it is possible to verify using a local equation of type u^{m_{i}(f)}v^{m_{j}(f)} + z^{2} = 0, that in both cases after the steps described above the total transform of {g = 0} is a normal crossing divisor, i.e. we do not have to blow up any other center, and the resolution procedure ends (cf. Orbanz [12].)

This ends the complete description of the exceptional set E and of all the normal bundles N_{E_{k}∩E_{k′} |E_{k}} (k ≠ k′).

3.5. The intersection of the strict transform St(g) of {g = 0} with E. Corresponding to the arrows with multiplicity (1) of the embedded resolution graph of the plane curve singularity {u^{m} + z^{2} = 0}, in the tower of surfaces above any A_{i}, there is exactly one, say Ẽ(A_{i}), which intersects the strict transform St(g). Let us denote this intersection by S_{i}^{m} = St(g) ∩ Ẽ(A_{i}).

First we will describe the position of S_{i}^{m} in Ẽ(A_{i}).

If m_{i}(f) is odd, the Ẽ(A_{i}) = X_{(2l_{i}+1)e_{i}−2x_{i}} = X_{0} = ℙ^{1} × ℙ^{1}. If by this identification π : Ẽ(A_{i}) → A_{i} corresponds to the first projection, and C_{0} and C_{1}, the intersection curves with the neighbor ruled surfaces in the same tower correspond to two fibers of the second projection, then S_{i}^{m} provides another section of π with S_{i}^{m} ∩ C_{0} = S_{i}^{m} ∩ C_{1} = ∅. Therefore S_{i}^{m} is another fiber of the second projection. Obviously its self-intersection is (S_{i}^{m})^{2} = 0 and S_{i}^{m} · f = 1.

The situation is slightly more complicated if m_{i}(f) = 2l_{i} is even.

Consider that moment of the resolution procedure when we finished the construction of the tower about A_{i}: we just created the last ruled surface X_{x_{i}−l_{i}e_{i}}, but we did not start the next tower above A_{i+1}. Consider the intersection points {P_{0}, …, P_{k}} of A_{i} with older exceptional curves A_{j} (j < i), and also all the intersection points {P′_{1}, …, P′_{l}} of A_{i} with all the other irreducible components of the total transform of {f = 0}, i.e. with the younger exceptional curves A_{j} (j > i) and with the strict transforms St_{j}. Then, similarly as above , we denote the collection of multiplicities m_{j}(f) of f along the components A_{j} (A_{j} · A_{i} ≠ 0, j > i) and St_{j} (St_{j} · A_{i} ≠ 0) by {m′_{t}}_{t=1}^{l} (such that the index corresponds to the index of the points P′_{t}). Then we are in the situation of (2.8) where X_{e} = X_{x_{i}−l_{i}e_{i}}. Using the notation of (2.8), the modified surface X_{e}^{m} is exactly the surface which is obtained from X_{x_{i}−l_{i}e_{i}} after we finish the resolution procedure, i.e. we construct all the neighbor towers as well.

Moreover, the intersection with the strict transform can be identified as follows. Assume that the x-chart ℂ_{x} of A_{i} contains all the points {P′_{t}}_{t=1}^{l}. Then the intersection S_{i} of the strict transform of g with X_{e} in the ℂ_{x} × ℙ^{1} chart is

{(x, [u_{0} : u_{1}]) : u_{0}^{2} = u_{1}^{2} ∏_{t=1}^{l}(x − x_{t})^{m′_{t}}} .

In particular, it is uniquely determined by the pair (A_{i}, {P′_{t}}_{t}) and from the numerical data. After we blow up X_{e} and we obtain X_{e}^{m}, denote the strict transform of S_{i} in X_{e}^{m} by S_{i}^{m} = St(g) ∩ X_{e}^{m}. Schematically, S_{i}^{m} is as in Figure 6 (but this is a “real picture”, which does not reflect exactly the “complex picture”). In the picture, the points P′_{i} with m′_{i} = 1 correspond exactly to the intersection points of A_{i} with the components of the strict transform of {f = 0}. Above these points S_{i}^{m} intersects transversally C_{1}^{m}, and these are the only intersection points of S_{i}^{m} and C_{1}^{m}.

Therefore,

S_{i}^{m} · C_{1}^{m} = #{strict transforms of f = 0 supported on A_{i}}
= #{m_{j}(f) = 1; D_{j} · A_{i} ≠ 0}

Since S_{i}^{m} → A_{i} is a double covering with branch locus exactly over the points P′_{j} with m′_{j} odd, one has:

• S_{i}^{m} · f = 2

• If each m_{j}′ is even (i.e. A_{i} has no neighbor with odd multiplicity) then S_{i}^{m} has two disjoint irreducible components, both isomorphic to A_{i}(≈ ℙ^{1}).

A_{i} C_{1}^{m} C_{0}^{m} P_{0}  P_{k} ⋯  P′_{j_{1}} m{′ \atop j_{1}} = 1  P′_{j_{2}} m{′ \atop j_{2}} = 2l^{j_{2}}  P′_{j_{3}} m′^{j_{3}} = 2l^{j_{3}} + 1 ≥ 3 −1 (1) −2 (1) −2 (1) −1 (1) −2 (1)  −1 −2 −2 (1) −3 (1) −1 (2) −2 (1) …  …  … S_{i}^{m}
Figure 6. S_{i}^{m}, the case m_{i}(f) = 2l_{i}

• If at least one m′_{j} is odd, then S_{i}^{m} is irreducible; its genus can be computed by an Euler characteristic argument (or Hurwitz’s formula, see [3] page 299):

genus(S_{i}^{m}) = {#\{j : m′_{j} odd\} − 2 \atop 2}.

Now we determine the self-intersection of S_{i}^{m}. The above diagram shows that S_{i}^{m} · C_{0}^{m} = 0 and also one can read all the intersections of S_{i}^{m} with the new exceptional divisors of X_{e}^{m}.

If m′_{j} is odd and ≥ 3, then above P′_{j} (i.e. in (π^{m})^{−1}(P′_{j})) we distinguish two irreducible curves F′_{j} and F_{j} defined by F_{j} · C_{i}^{m} = 1, F′_{j} · F_{j} = 1.

Notice that F′_{j} is the unique curve with multiplicity two in the divisor (π^{m})^{∗}(P′_{j}) and S_{i}^{m} intersects the fiber (π^{m})^{−1}(P′_{j}) exactly along F′_{j}. Therefore S_{i}^{m} · F′_{j} = 1. We invite the reader to verify that in Pic (X_{e}^{m}) one has

S_{i}^{m} = 2C_{1}^{m} + ∑ F_{j}, (sum over m′_{j} odd ≥ 3).

Therefore,

(S_{i}^{m})^{2} = (2C_{1}^{m} + ∑ F_{j})^{2}
= 4(C_{i}^{m})^{2} + 4M_{i} − 2M_{i} = 4(C_{i}^{m})^{2} + 2M_{i},

where M_{i} = #{j : m′_{j} odd, m′_{j} ≥ 3}.

If m_{i}(f) is even, and there is no D_{i} with D_{i} · A_{i} ≠ 0 with odd multiplicity, then S_{i}^{m} has two disjoint components, each ≈ C_{1}^{m} ≈ ℙ^{1} and ≡ C_{1}^{m} in Pic (X_{e}^{m}).

3.6. The self-intersections E_{k}^{2}. Consider all the compact irreducible exceptional divisors {E_{k}}_{k} of the resolution ϕ̃. In this paragraph we determine the “selfintersections” E_{k}^{2} := OX̃(E_{k})|E_{k} ∈ P ic(E_{k}). First notice that in the previous discussions we have determined completely the divisor (g ∘ ϕ̃) = St(g) + ∑_{l} m_{l}E_{l} (where the sum is over the compact irreducible exceptional divisors). But, for any k, (g ∘ ϕ̃) · E_{k} = 0 in P ic(E_{k}), therefore:

m_{k}E_{k}^{2} = (−St(g) − ∑_{l≠k} m_{l}E_{l})E_{k}

in P ic(E_{k}). But all the intersections St(g)E_{k} and E_{l}E_{k} (l ≠ k) are determined above, hence E_{k}^{2} follows.

3.7. The resolution of ({f(x, y) + z^{2} = 0}, 0). Notice that ϕ̃ : X̃ → ℂ^{3} induces a map ϕ̃: St(g) → {g = 0}, which is a resolution of the normal surface singularity {g = 0}. Its exceptional curve is exactly ⋃_{i} S_{i}^{m}. In this subsection we determine the self-intersections of the irreducible components of ⋃_{i} S_{i}^{m}, and their combinatorics. In particular, we re-obtain the dual resolution graph of this resolution (which was known cf. [6, 5, 1], see also [9]). The details are left to the reader.

We distinguish several cases.

If m_{i}(f) is odd, then π : S_{i}^{m} → A_{i} is an isomorphism, hence S_{i}^{m} is rational. Above each intersection point A_{i} ∩ A_{j} we have exactly one intersection point S_{i}^{m} ∩ S_{j}^{m}. The self-intersection of S_{i}^{m} in St(g) is e_{i}/2 where e_{i} = A_{i}^{2}.

If m_{i}(f) is even, and any D_{j} with D_{j} · A_{i} ≠ 0 has even multiplicity, then S_{i}^{m} has two irreducible components, each isomorphic to A_{i} = ℙ^{1}. In this case the selfintersection of each component in St(g) is e_{i}. Above an intersection point A_{i} ∩ A_{j} we have exactly two intersection points of S_{i}^{m} ∩ S_{j}^{m}.

If m_{i}(f) is even, and at least one of the neighbors (including the strict transform components) has odd multiplicity m_{j}(f), then S_{i}^{m} is an irreducible curve with genus = (λ − 2)/2 where λ is the number of odd neighbors (including the strict transform components) (cf. also with 3.5). Its self-intersection is 2e_{i}. Above an intersection point A_{i} ∩ D_{j} with m_{j}(f) even, there are two intersection points; if m_{j}(f) is odd, then only one intersection point of S_{i}^{m} ∩ S_{j}^{m}.

This determines completely the dual resolution graph of {f(x, y) + z^{2} = 0} from the embedded resolution graph of f. If we start with the minimal embedded resolution of f which has the additional property that there are no neighbors D_{i} and D_{j}, both with odd multiplicity, then the constructed resolution St(g) → {g = 0} is exactly the canonical resolution of {g = 0} (cf. [6]).

The above statements about the self-intersections of the components of S_{i}^{m} in St(g) can be obtained by the following “triple point formula” as well.

Let h : (ℂ^{3}, 0) → (ℂ, 0) be the germ of an analytic function and let (h) be the divisor of h ∘ ϕ̃. We assume that (h) is a normal crossing divisor. We write (h) = ∑_{D} m_{D}D where the sum runs over the irreducible exceptional divisors of ϕ̃ and the irreducible components of the strict transform of h. Let C be a compact curve determined by the intersection of two components D_{1} and D_{2}. Then

m_{D_{1}}(C^{2} in D_{2}) + m_{D_{2}}(C^{2} in D_{1}) + ∑ m_{D} = 0

where the last sum is over the triple points D ∩ D_{1} ∩ D_{2}. (For a proof, see e.g. [13].)

−1 −2 −2 −1 0 0 0 −1  −2 −1 −3 −10 0 0 0 0 −1 −2 −2 −2 1 −1 0 0 −1  1 −1 0 0 −1 1 −3 0  0 +2 −3 0 X_{3}^{m}  (2) E^{m}(A_{1}) (0)  E^{m}(A_{2}) (0)  E^{m}(A_{3}) (0) X_{0}  (2) X_{1}  (4) X_{2}^{m}  (6)  X_{1}  (3) X_{0}  (6) X_{1}  (2) 0 0  0

Figure 7. The exceptional divisor, case x^{2} + y^{3} + z^{2}

In our case, in order to obtain the self-intersections of S_{i}^{m}’s in St(g), we apply the above relation for h = g = f + z^{2}.

3.8. Example. Assume that f(x, y) = x^{2} + y^{3}. We start with the following resolution of f.

[figure: (1); (2); (6); (3); −3; −1; −2; A_{1}; A_{2}; A_{3}]

We fix the order A_{1}, A_{2}, A_{3} as it is indicated above. Then x_{1} = 0, e_{1} = −3; x_{2} = −1, e_{2} = −1; x_{3} = −3, e_{3} = −2. For the components of E, See Figure 7.

The dual resolution graph of ⋃_{i} S_{i}^{m} ⊂ St(g) is

[figure: −3; −3; −2; −1; S_{1}^{m}; S_{2}^{m}; S_{3}^{m}]

References

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3. R. Hartshorne Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer Verlag 1977.

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6. H. B. Laufer, On normal two-dimensional double point singularities, Israel Jornal of Math., 31, No. 3-4 (1978), 315-334.

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8. , Quasi-ordinary singularities of surfaces in ℂ^{3}, Singularities, Proc. Symp. Pure Math., 40), Amer. Math. Soc. Providence 1983, Part 2, 161–171.

9. A. Némethi, The signature of f(x, y) + z^{n}, Proceedings of Real and Complex Singularities, (C.T.C Wall’s 60th birthday meeting), Liverpool (England), August 1996; London Math. Soc. Lecture Notes Series, 263 (1999), 131-149.

10. I. Ono and K. Watanabe, On the singularity of z^{p} + y^{q} + x^{pq} = 0, Sci. Rep. Tokyo Kyoika Daigaku Sect. A, 12 (1974), 123-128.

11. P. Orlik and Ph. Wagreich, Isolated singularities of algebraic surfaces with ℂ^{∗} action, Ann. of Math., (2) 93 (1971), 205-228.

12. U. Orbanz, Enbedded resolution of algebraic surfaces after Abhyankar (characteristic 0), Lecture Notes in Math., 1101, Springer, 1984.

13. U. Persson, On Degenerations of Algebraic Surfaces, Mem. Amer. Math. Soc. 189 (Vol. 11), 1977.

E-mail address: cban@math.ohio-state.edu
E-mail address: mcewan@math.ohio-state.edu
E-mail address: nemethi@math.ohio-state.edu

Department of Mathematics, Ohio State University