Homology planes and algebraic curves

Tammo tom Dieck and Ted Petrie

1. Introduction and results

This paper is concerned with the construction of acyclic affine surfaces from plane algebraic curves.

Surfaces will always be connnected, non-singular, quasi-projective, algebraic surfaces over the complex numbers. Let R be a subring of the rational numbers. A surface V is called R–homology plane if H_{i}(X; R) = 0 for i > 0. In the case R = ZZ we simply refer to this as a homology plane.

We investigate homology planes via their compactifications. The compactifications give raise to an algorithmic construction of surfaces from curves in the projective plane IP^{2}. If X is a projective surface and C ⊂ X a curve we call (X, C) or X a compactification of any surface V which is isomorphic to the complement X \ C.

Ramanujam produced the first example of a homology plane. (See Ramanujam [1971]). His homology plane was in fact contractible and produced a counterexample to the conjecture that a smooth contractible affine surface over the complex numbers was the standard plane. Any smooth affine variety (over the reals) is the interior of a smooth manifold with boundary. In the case of a homology plane, it follows that the boundary is a homology sphere. When the homology plane is contractible, this observation gives homology planes bounding contractible manifolds — a point of interest in topology. It wasn’t until 1987 that other homology planes were found. Gurjar-Miyanishi (see Gurjar – Miyanishi [1987]) produced all homology planes of logarithmic Kodaira dimension 1. In this paper they ask whether there are an infinite number of contractible homology planes of Kodaira dimension 2. We announced the first affirmative solution to this problem in tom Dieck – Petrie [1989]. Among the results of this paper are the details of the announcement. Infinite families of homology planes of Kodaira dimension 2 result from the main theorems A and B here. See (3.20) and (3.21).

Another application of the main results here is the production of homology planes which have non trivial finite order automorphisms. These planes and finite order automorphisms produce counterexamples to the conjecture Petrie [1990] that non trivial homology planes have no finite order automorphisms. See tom Dieck [???] The contractible homology planes of Kodaira dimension 1 have no finite order automorphisms.

There is recent interest in the higher dimensional analogs of homology planes, i. e. acyclic affine varieties of arbitrary dimension. Some are obtained as products of homology planes with C^{k}. Others are produced as hypersurfaces in C^{n} in tom Dieck – Petrie [1990]. These analogs are related to the Abhyankar-Moh Problem: Is any hypersurface in C^{n} which is isomorphic to C^{n−1} actually ambiently isomorphic to C^{n−1}? The point is that it is possible to produce acyclic hypersurfaces in C^{n} but the question remains whether they are isomorphic to C^{n−1} and therefore potential negative responses to the problem. One related question is: For which homology planes X can X × C^{k} be a hypersurface in C^{n}? See tom Dieck – Petrie [1990] for the case n = 3. Perhaps the explicit homology planes produced from Theorems A and B will yield answers to these problems.

In this paper we give a construction of all homology planes. This is the content of Theorem A. The construction produces other affine surfaces as well. Theorem B and Theorem 3.13 tell which among these are homology planes and the Determinant Algorithm 4.27 gives an effective method of applying Theorems B and 3.13. The results of the first 4 sections deal with the existence of homology planes. The constructions begin with a curve in a minimal projective surface (which can be the projective plane). If there is one homology plane associated to such a curve, there are infinitly many distinct homology planes associated with the same curve. It can even happen that the same homology plane can be associated with different curves. Any classification of homology planes must take this into account. In subsequent notes we illustrate this point by using e. g. the group of birational automorphisms of the projective plane to produce isomorphisms between some of the homology planes constructed here and elsewhere.

Our first theorem explains the construction of homology planes from curves. In order to state it we need some notation.

By an expansion p : X → Y of Y we mean a composition

p : X = X_{k+1} p_{k}_{\overline{}}X_{k} → … → X_{1} p_{0}_{\overline{}}X_{0} = Y,

where each morphism p_{j} blows up a single point. Blowing up a single point is also called a σ–process. An expansion p : X → Y of Y is called a contraction of X. The exceptional set of any expansion p : X → Y is

Σ_{p} = Σ(p) = {y ∈ Y | p^{−1}(y) is not a point}.

The exceptional divisors of p are the components of p^{−1}(x) for x ∈ Σ_{p}. Let E = E(p) denote the set of exceptional divisors.

We now explain the basic setting for the whole paper. Let D = D_{1} ∪ … ∪ D_{n} ⊂ Y be a curve in a minimal rational projective surface Y such that the D_{i} are rational and the (irreducible) components of D. There is a unique minimal expansion π : X → Y such that π^{−1}(D) = C has normal crossings (embedded resolution of singularities). Minimal means that any other expansion with this property factors over π.

Choose a partition E = E_{0} ∐ E_{1} and define d : E → {0, 1} by d^{−1}(i) = E_{i}. Let D′ = p^{#}D denote the proper transform of D. Consider the curve (= reduced effective divisor, additive notation) D(d) in X

(1.1) D(d) := D′ + ∑_{E∈E_{1}} E ⊂ p^{−1}(D) = D′ + ∑_{E∈E} E.

Then D(d) is a normal crossing curve and has a dual graph ΓD(d) with vertex set Γ_{0}D(d) and edge set Γ_{1}D(d). If Γ is a graph we set s(Γ) = 1 − χ(Γ), where χ denotes the Euler characteristic. If Γ is connected, then s(Γ) is the number of its (independent) cycles. We refer to the beginning of section 2 for our terminology relating to graphs. In the case of Γ = ΓD(d) we set

(1.2) s(d) = 1 − χ(ΓD(d)), r(d) = |E_{0}|.

The function d : E → {0, 1} is called a selection function for D, if the following holds:

(1.3) \begin{matrix}(1) & ΓD(d) is connected \\ (2) & r(d) + s(d) = n − b_{2}(Y) \end{matrix} (b_{2} = second Betti number).

Affine varieties have a connected compactification divisor (1.3.1). The equality (1.3.2) is basic for varieties of general type. This fact will be explained in the proof of Proposition (2.3). The term ‘selection function’ simply refers to the fact that one has in general a choice for such a function.

We want to cut the cycles of ΓD(d). We call Φ ⊂ Γ_{1}D(d) a cutting set for (D, d) if the subgraph with vertex set Γ_{0}D(d) and edge set Γ_{1}D(d) \ Φ is a tree. (Thus |Φ| = s(d), provided (1.3.1) holds. The case Φ = ∅ is not excluded.)

Given (D, d, Φ) we let Bl(D, d, Φ) denote the set (of isomorphism classes) of expansions p : Z(p) → X with the following properties (1.4). According to our terminology in section 2, the set Φ is a set of points in X, so that the statement (1.4.1) makes sense.

(1.4) \begin{matrix}(1) & Σ_{p} = Φ. \\ (2) & For each x ∈ Φ the graph of A(x) ∪ B(x) ∪ p^{−1}(x) is a linear \\ & tree. Here A(x) and B(x) are the curves which intersect in x. \\ (3) & Each curve p^{−1}(x), x ∈ Φ, contains a unique component \\ & E_{0}(x) with self-intersection −1 (called (−1)–curve). \end{matrix}

Expansions with these properties are referred to as standard. Late in this paper these expansions are called subdivisional. This terminology is taken from Fujita [1982], (3.4) where it is proved that a Q-homology plane is an affine surface. Let

(1.5) B(p) = p^{−1}(D(d)) − ∑_{x∈Φ} E_{0}(x) ⊂ Z(p).

The family of surfaces (1.6) V (p) = Z(p) \ B(p), p ∈ Bl(D, d, Φ)

is called the tower of surfaces belonging to (D, d, Φ). If Φ is empty, the tower consists of the single element X \ D(d). Open surfaces V have a logarithmic Kodaira dimension κ̄(V) ∈ {−∞, 0, 1, 2}. It is known that homology planes V with κ̄(V) ∈ {−∞, 0} dont’t exist.

Theorem A. Let V be a homology plane of general type. Then there exists a rational curve D in some minimal rational surface Y, a selection function d for D and a cutting set Φ for (D, d) such that V is isomorphic to V (p) for some p ∈ Bl(D, d, Φ).

In fact, Y can always be chosen to be IP^{2}. But for some considerations other minimal surfaces Y are more appropriate.

The homology planes with κ̄ = 1 have been described by Gurjar – Miyanishi [1987]. Therefore we concentrate on surfaces with κ̄ = 2, called of general type.

General considerations suggest that an algorithm like Theorem A exists for construction of homology planes. By compactification a homology plane V is Z \ B where (Z, B) is a compactification of V. The exceptional divisiors lying on B can be blown down giving a new surface X and a curve Y such that V = X \Y. Further considerations show in fact X can be a minimal rational surface and Y rational. Still given a rational minimal surface X and a rational curve Y ⊂ X, it is not at all obvious if one can reverse this process and produce a homology plane. Theorem A and its companions Theorems B and 4.27 give an effective algorithm for deciding this.

By Poincaré duality the irreducible components of D or D(d) define elements in H^{2}(Y) or H^{2}(X). This is explained in the beginning of section 3. Using this fact we state:

Addendum to Theorem A. Under the hypothesis of Theorem A the irreducible components of D (resp. of D(d)) generate H^{2}(Y) (resp. H^{2}(X)).

The next theorem tells us which surfaces V (p) in a tower are homology planes. Suppose the edge x ∈ Φ is an intersection of the components A(x), B(x) of D(d), denoted ∂(x) = {A(x), B(x)}. Let m_{p}(A(x)) denote the multiplicity of E_{0}(x) in the total transform p^{∗}A(x) (pullback of A(x) as Cartier divisor, compare Definition (3.6)). Then m_{p}(A(x)) and m_{p}(B(x)) are coprime positive integers. Let ZZ(∂x) denote the free abelian group with basis {A(x), B(x)}. Let

m_{p} ∈ ⨁_{x∈Φ} ZZ(∂x)

be the element with x–component m_{p}(A(x))A(x) + m_{p}(B(x))B(x). A multiplicity function m for (D, d, Φ) assigns to each x ∈ Φ a pair of coprime integers (m(A(x)), m(B(x)). We also denote by m the corresponding element in ⨁_{x∈Φ} ZZ(∂x).

Theorem B. Let D be a curve with rational components in a minimal rational projective surface Y. Let d be a selection function for D such that the components of D(d) generate H^{2}(X). Let Φ ≠ ∅ be a cutting set for (D, d). Then there exists a multilinear function

Δ = Δ(D, d, Φ) : ⨁_{x∈Φ} ZZ(∂x) → ZZ

such that:

(1) V (p), p ∈ Bl(D, d, Φ), is a Q–homology plane if and only if Δ(m_{p}) ≠ 0.

(2) V (p) is a ZZ–homology plane if and only if Δ(m_{p}) = ±1.

(3) If Δ(m_{p}) ≠ 0, then |H^{2}(V (p); ZZ)| = |H_{1}(V; ZZ)| = |Δ(m_{p})|.

The function Δ(D, d, Φ) will be called the discriminant of (D, d, Φ). The discriminant is defined up to sign.

The multiplicity functions can be used to parametrize elements in Bl(D, d, Φ).

(1.7) Proposition. Using the notation set up so far wecan state: The assignment p ↦ m_{p} sets up a bijection between Bl(D, d, Φ) and the set Mul(D, d, Φ) of multiplicity functions for (D, d, Φ). □

The following elemtary algebraic remarks use (1.7). By linearity, Theorem B determines |Δ(m)| for each m ∈ ⨁ ZZ(∂x). Two bilinear maps Δ_{1} and Δ_{2} which have the same absolute value |Δ_{1}(m)| = |Δ_{2}(m)| for all arguments m are equal up to sign. Therefore there exists up to sign at most one bilinear map Δ which has the properties stated in Theorem B.

By elementary algebra we see that if Δ assumes the value 1, then it does so for infinitely many arguments. Therefore if a tower contains homology planes at all, then it contains an infinite family.

The proof of Theorem B will show that the discriminant is effectively computable. We do not claim that members of a tower are always surfaces of general type. There are usually some degenerate cases is a tower. We shall see examples later. Different towers may contain isomorphic varieties.

Theorems A is proved in section 2; Theorem B is proved in section 3. In section 4 we prove the Addendum to Theorem A and Proposition (1.7) In section 5 we compare towers of different curves via birational morphisms and give several applications. For an annoucement and further information see tom Dieck – Petrie [1989].

2. Towers of surfaces

This section is devoted to the proof of Theorem A.

We begin by introducing some terminology. A graph Γ = (Γ_{0}, Γ_{1}) consists of two sets Γ_{0}, the set of vertices, and Γ_{1}, the set of edges, and a map ∂ which assigns to each x ∈ Γ_{1} a non empty set ∂x ⊂ Γ_{0} of at most two elements. The geometric realization of Γ is a one-dimensional CW–complex with 0–skeleton Γ_{0} and with a 1–simplex corresponding to each x ∈ Γ_{1} attached to ∂x in the obvious manner. We do not distinguish notationally between Γ and its geometric realization and apply a geometric terminology to Γ, like: cycle, subdivision, Euler characteristic, the edge x connects the points in ∂x etc.

A curve C ⊂ X is said to have normal crossings if its irreducible components are non-singular, two components intersect in at most one point and such a point is a transverse intersection, and there are no triple points. A normal crossing curve C has a weighted dual graph (ΓC, w). The graph ΓC = (Γ_{0}C, Γ_{1}C) has as vertex set Γ_{0}C the set of irreducible components of C and as edge set Γ_{1}C the set of double points. An edge connects the vertices which intersect in it, i. e. if A∩B = {x}, then ∂(x) = {A, B}. Therefore Γ_{1}C is also a set of points in X, but the Γ–notation always referres to the graph. The weight function w : Γ_{0}C → ZZ associates to A ∈ Γ_{0}C its self-intersection A · A in X. The intersection matrix is Γ_{0}C × Γ_{0}C → ZZ, (A, B) ↦ A · B and its determinant is denoted det(ΓC, w).

(2.1) Proposition. Suppose C is a normal crossing curve in the rational projective surface X. Then the following are equivalent:

(1) V = X \ C is a Q–homology plane.

(2) The components of C are rational curves and ΓC is a tree. The inclusion C ⊂ X induces an isomorphism H_{2}(C; Q) → H_{2}(X; Q).

Proof. We use co-homology with coefficients in Q. From the homological structure of topological surfaces one deduces easily: H_{1}(C) = 0 if and only if the components of C are 2–spheres and ΓC is a tree.

We combine the exact homology sequence of (X, C) with Poincaré duality H^{4−i}(X, C) ≅ H_{i}(V) and H^{∗}(A) ≅ H_{∗}(A) for the appropriate spaces A.

(1) ⟹ (2) By hypothesis H_{i}(X, C) ≅ H^{4−i}(V) ≅ 0 for 0 ≤ i ≤ 3. The exact sequence of (X, C) and H_{1}(X) ≅ 0 implies H_{1}(C) ≅ 0 and the isomorphism H_{2}(C) ≅ H_{2}(X).

(2) ⟹ (1) From the hypothesis and the remark at the beginning H_{1}(C) ≅ 0 and H_{3}(X) ≅ 0. The isomorphism H_{2}(C) ≅ H_{2}(X) and the exact sequence of (X, C) imply H_{2}(X, C) ≅ H_{3}(X, C) ≅ 0 and hence H_{1}(V) ≅ H_{2}(V) ≅ 0. Since C is connected we obtain H_{3}(V) ≅ H_{4}(V) ≅ 0 from the exact sequence. □

Let V be a homology plane of general type. By the rationality theorem of Gurjar – Shastri [1989], there exists a rational projective surface X_{1} and a curve D_{1} ⊂ X_{1} such that V is isomorphic to X_{1} \ D_{1}. By embedded resolution of singularities (Barth – Peters – Van de Ven [1984], II(7.2))we can assume that D_{1} has normal crossings. Let p_{1} : X_{1} → Y be a contraction to a minimal rational surface and set D = p_{1}(D_{1}). We can expand further by p_{2} : Z → X_{1} such that (p_{1}p_{2})^{−1}D has normal crossings. Let π = π_{D} : X = X_{D} → Y denote the minimal expansion such that π^{−1}(D) = C has normal crossings. By minimality of π we have a factorization p_{1}p_{2} = πp with an expansion p : Z → X. Let B = p_{2}^{−1}(D_{1}). We have V ≅ Z \ B by construction. We collect some properties of (p, B). For the convenience of the reader we display the data above in the following diagram.

B
∩
Z

[figure: p_{2}; p]
D_{1} ⊂ X_{1}  X ⊃ C

[figure: p_{1}; π]
Y
∪
D

(2.2) Proposition. Let V be a Q–homology plane. The pair (p, B) has the following properties:

(1) πp(B) = D.

(2) Σ(πp) ⊂ D.

(3) ΓB is a tree.

(4) |Γ_{0}(πp)^{−1}D| − |Γ_{0}B| = |Γ_{0}D| − b_{2}(Y).

Proof. The equality (1) holds by construction.

(2) If x ∈ Σ(πp) \ D, then p^{−1}(x) ⊂ Z \ B = V and at least one component of p^{−1}(x) is a (−1)–curve E. Contract E to obtain V′. Then H^{2}(V) ≅ ZZ ⊕ H^{2}(V′) contradicting H^{2}(V; Q) = 0.

(3) is given by (2.1).

(4) We have |Γ_{0}(πp)^{−1}D| − |Γ_{0}D| = b_{2}(Z) − b_{2}(Y) as a general property of expansions and |Γ_{0}B| = b_{2}(Z) by (2.1). This yields the result. □

The curve p(B) is contained in π^{−1}(D) but may be different from it. There is a function d : E(π) → {0, 1} such that p(B) = D(d), see (1.1).

(2.3) Proposition. d is a selection function for D.

Proof. We have to verify (1.3). Since V is a Q–homology plane, ΓB is a tree, so the curve p(B) = D(d) is connected; hence so is p(B) = D(d) and therefore its weighted dual graph ΓD(d).

In order to prove (1.3.2) we have to analyze in detail the passage from X to Z.

We write p as a composition of σ–processes

p : Z = Z_{k+1} ^{p_{k} \overline{}}Z_{k} → … → Z_{1} ^{p_{0} \overline{}}Z_{0} = X.

The σ–process p_{j} blows up a point x_{j} to E_{j}(x_{j}).

Let D_{j} ⊂ Z_{j} denote the total transform of D(d) =: D_{0} and B_{j} ⊂ Z_{j} the image of B. Then B_{j} ⊂ D_{j}. The graphs ΓD_{j} and ΓD_{j+1} are related in either one of the following ways:

(2.4) The process p_{j} blows up a double point x_{j} of D_{j} to E(x_{j}). Then ΓD_{j+1} is obtained from ΓD_{j} by subdividing the edge x_{j} ∈ Γ_{1}D_{j}. This p_{j} is called subdivisional expansion.

[figure: ⟹; A; B; A; B; E(x_{j}); ⟹; ΓD_{j}; ΓD_{j+1}; x_{j}]

(2.5) The process p_{j} blows up a regular point x_{j} of D_{j} in the component A. Then ΓD_{j+1} is obtained from ΓD_{j} by attaching a new edge to ΓD_{j} at the vertex A ∈ Γ_{0}D_{j}. This p_{j} is called a sprouting expansion.

[figure: ⟹; ΓD_{j}; ⟹; ΓD_{j+1}; A; A; E(x_{j})]

The graphs ΓB_{j} and ΓB_{j+1} are related in either one of the following ways:

(2.6) B_{j+1} is the total transform of B_{j}. In this case ΓB_{j+1} arises from ΓB_{j} either by subdividing as in (2.4) or by sprouting as in (2.5).

(2.7) B_{j+1} is the proper transform of B_{j}. If p_{j} is subdivisional, then ΓB_{j+1} is obtained fromΓB_{j} by removing the edge x_{j} ∈ Γ_{1}B_{j}. If p_{j} is sprouting, then ΓB_{j} and ΓB_{j+1} are combinatorially isomorphic.

Of course, passage from B_{j} to B_{j+1} changes the weights.

The geometric effect in (2.7) is the following. If p_{j} in (2.7) is subdivisional, then the edge x_{j} ∈ Γ_{1}B_{j} must be contained in a cycle, for otherwise ΓB_{j+1} and hence ΓB would not be connected. Therefore in this case s(ΓB_{j+1}) = s(ΓB_{j})−1, i. e. the number of cycles decreases by one (see the notation introduced before (1.2)). If p_{j} in (2.7) is sprouting, then ΓB_{j+1} and ΓB_{j} are homeomorphic. This shows:

Since ΓD(d) has s(d) cycles and ΓB is a tree there are exactly s(d) subdivisional expansions (2.7).

The total number w = |Γ_{0}p^{−1}D(d)| − Γ_{0}B| of expansions (2.7) is counted as follows.

|Γ_{0}(πp)^{−1}D| − |Γ_{0}B| =
|Γ_{0}(πp)^{−1}D| − |Γ_{0}p^{−1}D(d)| + |Γ_{0}p^{−1}D(d)| − |Γ_{0}B| =
|Γ_{0}π^{−1}D| − |Γ_{0}D(d)| + |Γ_{0}p^{−1}D(d)| − |Γ_{0}B| =
r(d) + w.

This uses the definition of r(d) in (1.2). From the string of equalities, (2.2.4) and the previous paragraph we obtain

|Γ_{0}D| − b_{2}(Y) = r(d) + w ≥ r(d) + s(d).

This is half of (1.3.2)

In order to prove equality we have to use the assumption that V is a surface of general type. For if we would not have equality, then a sprouting expansion p_{j} would occur in (2.7). In that case E(x_{j}) intersects B_{j+1} in a single point and E_{j} = E(x_{j})\E(x_{j})∩B_{j+1} is a curve in Z_{j+1}\B_{j+1} which is isomorphic to the affine line C. The proper transform of this curve in V = Z \ B is also isomorphic to C. This contradicts a result of Zaĭdenberg [1988] and of Miyanishi – Tsunoda [1990], which asserts that a homology plane of general type does not contain such a curve. □

We use the discussion in the proof of (2.3) to state another fact. We call the expansion p : Z → X minimal if the following holds. Let p_{1} : Z_{1} → X be an expansion and B_{1} ⊂ Z_{1} a curve such that (2.2) holds for (p_{1}, B_{1}) in place of (p, B). Suppose moreover that ω : Z → Z_{1} is an expansion such that ω^{−1}(B_{1}) = B and p_{1}ω = p. If in any such situation ω is an isomorphism, then (p, B) is called minimal.

(2.8) Proposition. If p : Z → X is minimal there are no sprouting expansions (2.6) and (2.7).

Proof. In the proof of (2.3), by verifying (1.3.2), we have already seen that there are no sprouting expansions (2.7). Suppose p_{j} is a sprouting expansion (2.6). Then E(x_{j}) ⊂ B_{j+1} is a (−1)–curve. It represents a vertex in Γ_{0}B_{j+1} which is not a branch point, i. e. is contained in at most two edges, and which is not contained in a cycle. One now shows by induction that ΓB_{r} for r ≥ j +1 contains a vertex with weigth −1 which is not a branch point and which is not contained in a cycle. The fact that B_{k+1} = B contains such a curve shows that p : Z → X is not minimal. □

Using (2.8) it is now easy to verify by induction on j:

(2.9) Proposition. If p : Z → X is minimal, then ΓD_{j} is a subdivision of ΓD_{i}, i < j, and ΓB_{j} is a subcomplex of ΓD_{j}. □

Proof of Theorem A. Let V be a homology plane of general type. In the preceding discussion we have already obtained the follwing data: A minimal rational surface Y, a rational curve D in Y, a selection function d for D, an expansion p : Z → X and a curve B ⊂ Z such that V ≅ Z \ B.

If (p, B) were not minimal we could pass to a minimal situation by suitable contractions. So let us assume that (p, B) is minimal. This is then perhaps no longer the expansion which appears in the diagram before (2.2), but (2.2) and (2.3) still hold and therefore also (2.8) and (2.9).

We claim that Σ_{p} is a cutting set for (D, d) and satisfies (1.4).

Property (1.4.2): This follows from (2.8).

Property (1.4.3): In geometric terms Σ_{p} ∈ Γ_{1}D(d) is precisely the set of those edges which are subdivided when we pass from ΓD(d) to ΓD_{k+1}. By minimality of (p, B), the curve B does not contain the (−1)–curves in p^{−}1(x), x ∈ Σ_{p}. If p^{−}1(x) contains more than one (−1)–curve, then ΓB would not be connected; the proof is similar to the proof of (2.8). Therefore (1.4.3) holds.

Property (1.4.1): The edge x_{j} ∈ Γ_{1}B_{j} of a subdivisional expansion (2.7) is contained in an edge of ΓD(d) and this edge is contained in a cycle of ΓD(d). Conversely, each edge Σ_{p} ⊂ Γ_{1}D(d) contains exactly one edge belonging to a sudivisional expansion (2.7). Since ΓB is a tree, Σ_{p} must be a cutting set.

A detailed description of expansions together with their weighted dual graphs will be given in section 4.

We add the following remark. Let P_{j} = {p_{j} | j ∈ J(x)} be the set of expansions whose exceptional divisor is mapped to x ∈ Σ_{p}. There is a unique expansion of type (2.7) in P_{j} and this is the one with the maximal j ∈ J(x). □

3. The discriminant

This section is devoted to the proof of Theorem B. We need a computation of H^{2}(V) in terms of multiplicities. Cohomology with integral coefficients will be used. We think of V as being one of the surfaces in a tower.

We begin by recalling some general facts. A curve A in a projective surface X defines an element ⟨A⟩ ∈ H^{2}(X). There are two ways of explaining this fact. Firstly, if A ⊂ X is an irreducible curve, then A is the continuous image under f : Ã → X of an oriented closed 2–manifold Ã. The fundamental class of Ã has an image z_{A} ∈ H_{2}(X) and ⟨A⟩ ∈ H^{2}(X) corresponds to z_{A} under Poincaré duality.

Secondly, the divisor A defines an element in the Picard group Pic(X) and ⟨A⟩ is its image under the first Chern class homomorphism Pic(X) → H^{2}(X).

If E is a set of irreducible curves in X we let ZZ(E) denote the free abelian group on the elements of E. We thus have a homomorphism

(3.1) α(E) : ZZ(E) → H^{2}(X)

which maps A ∈ E to ⟨A⟩.

The following Proposition collects some general facts about expansions.

(3.2) Proposition. Let p : X → Y be an expansion with set of exceptional divisors E_{p}. Then the following holds:

(1) If B ⊂ Y is any curve and B′ = p^{#}B denotes its proper transform, then p^{∗}⟨B⟩ − ⟨B′⟩ = α(E_{p})(y) for a unique element m(B) := y ∈ ZZ(E_{p}).

(2) The element m(B), written as a linear combination of the basis elements E ∈ E, has non-negative coefficients.

(3) The homomorphism

⟨p^{∗}, α(E_{p})⟩ : H^{2}(Y) ⊕ ZZ(E_{p}) → H^{2}(X), (x, y) ↦ p^{∗}(x) + α(y)

is an isomorphism.

Proof. Suppose p is a σ–process. Then (3) follows from the Mayer-Vietoris-sequence in cohomology by using the fact that a σ–process is the connected sum with a projective plane. Using this, (1) and (2) follow from Hartshorne [1977], V(3.6). The general case is now a straightforward induction on the number of σ–processes involved in the expansion. □

By linear extension, the elements m(B) in (3.2.1) yield a homomorphism

(3.3) m : ZZ(Γ_{0}D) → ZZ(E_{p})

for any curve D ⊂ Y and any expansion p : X → Y. We also use the elements m(A) in the next definition.

(3.4) Definition. If A ∈ Γ_{0}D and m(A) = ∑_{E∈E_{p}} m(A, E)E, then m(A, E) ∈ ZZ

is called the multiplicity of E in the total transform p^{∗}⟨A⟩ of A.

(3.5) Proposition. Let p : X → Y be an expansion with set of exceptional divisors E_{p}. Let A ⊂ Y be an irreducible curve. The multiplicity m(A, E), E ∈ E_{p}, is non-zero if and only if p(E) ⊂ A.

Proof. This follows by induction on the number of σ–processes from Hartshorne [1977], V(3.5.2),(3.6). □

After this recollection we turn to the cohomological study of complements of curves.

(3.6) Assumptions. Let C be a curve in the rational projective surface X with the following properties:

(1) The components of C are rational.
(2) H_{1}(C) = 0.
(3) H_{2}(C) is free abelian. A basis is given by the fundamental classes of

the components of C.

(4) |Γ_{0}C| = b_{2}(X). (Γ_{0}C = set of components).

We do not assume that C has normal crossings.

(3.7) Proposition. Suppose (X, C) satisfies (3.6). Let j : V = X \ C → X denote the inclusion. Then the following holds:

(1) The sequence

ZZ(Γ_{0}C) α(Γ_{0}C)_{\overline{}}H^{2}(X) j^{∗}_{\overline{}}H^{2}(V) \overline{}0

is exact.

(2) V is a ZZ– (resp. Q–) homology plane if and only if H^{2}(V) is zero (resp. finite).

Proof. Consider the diagram

H^{2}(V) j^{∗}_{\overline{}}H^{2}(X)

[figure: P; P]
[figure: H_{1}(C); H_{2}(X, C); H_{2}(X); i_{∗}; H_{2}(C).]

The maps P are Poincaré duality isomorphisms. The bottom row shows part of the homology sequence of (X, C). By (3.6.3) we can identify H_{2}(C) = ZZ(Γ_{0}C) and then P^{−1} ∘ i_{∗} becomes α(Γ_{0}C). Thus (3.7.1) follows.

The exact homology sequence of the pair (X, C), the assumptions (3.6) and the fact that H_{1}(X) ≅ H_{3}(X) ≅ 0 for a rational projective surface imply: The groups H_{i}(X, C) are zero for 0 ≤ i ≤ 3 if and only if H_{2}(C) → H_{2}(X) is an isomorphism. By Poincaré duality this implies: H̃^{∗}(V) = 0 if and only if H_{2}(C) → H_{2}(X) is an isomorphism. This holds for integral or rational coefficients.

Since H_{2}(C) and H_{2}(X) have the same rank (3.6.4), the assertions of the last paragraph are easily translated into the statement (3.7.2). □

We set L(C) = image α(Γ_{0}C) and obtain under the hypothesis of (3.7) an isomorphism H^{2}(V) ≅ H^{2}(X)/L(C).

We consider a pair (X, C) that arises from (Y, D) in a particular way:

(3.8) Data and assumptions on (p : X → Y, D, C).

(1) p : X → Y is an expansion.
(2) D ⊂ Y is a curve with rational components.
(3) Σ_{p} ⊂ D.
(4) E := E(p) = E_{0} ∪ E_{1} disjoint union.
(5) C = D′ + ∑_{E∈E_{1}} E.  (D′ = p^{#}D)
(6) C satisfies (3.6).
(7) α(Γ_{0}D) is surjective and has kernel R.
(8) m_{0} : ZZ(Γ_{0}D) → ZZ(E_{0}) is the composition of m in (3.5) with the

canonical projection ZZ(E) → ZZ(E_{0}). (3.9) Proposition. Suppose data and assumptions (3.8) are given. Then

H^{2}(V) ≅ H^{2}(X)/L(C) ≅ ZZ(E_{0})/m_{0}(R).

Proof. Diagram chasing in the following commutative diagram.

H^{2}(X)  ^{\overline{}⊃}  L(C)

[figure: ⟨p^{∗}, α(E)⟩; H^{2}(Y) ⊕ ZZ(E); α(Γ_{0}(D)) ⊕ id; α(Γ_{0}D′ ∪ E_{1})]
ZZ(Γ_{0}D) ⊕ ZZ(E)  ^{\overline{}δ}  ZZ(Γ_{0}D′) ⊕ ZZ(E_{1})
R

The map δ is defined as δ(x′, y) = (x, y − m(x)). □

We express (3.9) more explicitly for the cases of interest. Let R_{1}, …, R_{k} be a basis for R = kernel α(Γ_{0}D), called relation basis. Then

k = |Γ_{0}D| − b_{2}(Y), b_{2}(Y) + |E| = b_{2}(X) = |Γ_{0}C| = |Γ_{0}D| + |E_{1}|, whence k = |E_{0}|. Consider the elements of Γ_{0}D as a basis for ZZ(Γ_{0}D). Then R_{j} is an integral linear combination

(3.10) R_{j} = ∑_{A∈Γ_{0}D} c_{j}(A)A, c_{j}(A) ∈ ZZ.

For A ∈ Γ_{0}D the element m_{0}(A), see (3.8.8), is an integral linear combination

(3.11) m_{0}(A) = ∑_{E∈E_{0}} m(A, E)E.

From (3.10) and (3.11) we obtain an equality of the following type

(3.12) m_{0}(R_{j}) = ∑_{E∈E_{0}} λ(j, E)E, λ(j, E) ∈ ZZ.

The coefficients a(j, E) constitute a (k, k)–matrix Λ = (λ(j, E)), called multiplicity relation matrix, or MR-matrix for short.

Now (3.9) can be reformulated as follows.

(3.13) Theorem. Suppose data and assumption (3.8) are given. Then V = X \ C is a Q–homology plane if and only if the MR-matrix Λ = (λ(j, E)) has non-zero determinant. If det Λ ≠ 0, then |H^{2}(V)| = | det Λ|. The surface V is a homology plane if and only if | det Λ| = 1. □

Proof of Theorem B. We apply (3.13) to (p : Z(p) → X, D(d), B(p)) for p ∈ Bl (D, d, Φ) in place of (p : X → Y, D, C). We have to check (3.8). We have assumed that α(Γ_{0}D(d)) is surjective. It only remains to check that B(p) satisfies (3.6). Part (3.6.1) holds by construction; (3.6.2) and (3.6.2) follow from (3.6.1) and because ΓB(p) is a tree.

In order to verify |Γ_{0}B(p)| = b_{2}(X) we note:

b_{2}(X) − b_{2}(Y) = |Γ_{0}π^{−1}D| − |Γ_{0}D|
= |Γ_{0}D(d)| + |d^{−1}(0)| − |Γ_{0}D|

and the definition of s(d) imply

b_{2}(X) = |Γ_{0}D(d)| − s(d).

Moreover we have

b_{2}(Z) = |Γ_{0}p^{−1}D(d)| − |Γ_{0}D(d)|
= |Γ_{0}B(p)| + |Φ| − |Γ_{0}D(d)|
= |Γ_{0}B(p)| + s(d) − |Γ_{0}D(d)|.

The two relations imply the desired equality.

The MR-matrix is again computed from (3.10) and (3.11) which read in this case

R_{j} = ∑_{A∈Γ_{0}D(d)} c_{j}(A)A

and

m_{0}(A) = ∑_{x∈Φ} m(A, E_{0}(x))E_{0}(x).

The MR-matrix therefore has the entries

(3.14) λ(j, x) = ∑_{A∈Γ_{0}D(d)} c_{j}(A)m(A, E_{0}(x)).

By (3.5) and (1.4.1) there are exactly two A ∈ Γ_{0}D(d) with m(A, E_{0}(x)) ≠ 0, namely the elements called called A(x) and B(x) before the statement of Theorem B. Thus m_{p}(A(x)) = m(A(x), E_{0}(x)) and similarly for B(x). From (3.14) we see now that the x–column of the MR-matrix Λ is a vector of the form

(c_{j}(A(x))m_{p}A(x) + c_{j}(B(x))m_{p}B(x)).

Since the c_{j}(A) only depend on the choice of the relation basis they are independent of p. The discriminant Δ(D, p, Φ) is now defined to be the determinant of Λ, viewed formally as a function in the variables (m_{p}A(x), m_{p}B(x)), x ∈ Φ. Statements (1) – (3) in Theorem B are finally a restatement of (3.13). □

Theorem (3.13) is more genral than Theorem B and allows calculation of the discriminant without passing to a normal crossing curve. An example of this type is given in (3.15)

(3.15) The Ramanujam tower. The first non-trivial example of a contractible homology plane was published by Ramanujam [1971]. The starting point was a curve D = R ∪ S in IP^{2}. It consists of a cuspidal cubic R ⊂ IP^{2} and a regular quadric S ⊂ IP^{2} with intersection pattern R · S = P + 5Q. Let p : Z(p) → IP^{2} be a standard expansion (1.4) with Σ_{p} = {P }. We consider the tower consisting of surfaces V (p) = Z(p) \ B(p). Then (Z(p), B(p)) satisfies (3.6) and (p : Z(p) → IP^{2}, D, B(p)) satisfies (3.8) with E_{0} the (−1)–curve E in p^{−1}(P). We have H^{2}(IP^{2}) = ZZ and α(Γ_{0}D) : aR + bS ↦ 3a − 2b is surjective with relation basis given by R_{1} = 2R−3S. The expansion p is determined by the multiplicities m(R, E), m(S, E) and m_{0}(R_{1}) = (2m(R, E) − 3m(S, E))E. Hence 2m(R, E) − 3m(S, E) is the value of the discriminant. The case m(R, E) = m(S, E) = 1 was the example given by Ramanujam. The surface with m(R, E) = 2, m(S, E) = 1 is isomorphic to an example in Gurjar – Miyanishi [1987].

In the systematic treatment (i. e. in the tower algorithm of Theorem B) we would first pass to the minimal resolution π : X → IP^{2} of D. The weighted dual graph of π^{−1}(D) = C is

[figure: −2; −1; −2; (c); −1; −2; −2; −2; −2; −3; −1; (b); S′; (a); R′]

It has a single cycle. A selection function must have constant value one. There are three cutting sets, consisting of the edges (a), (b), or (c).

The (−1)–curve S′ can be contracted. Then π^{−1}(D) is a curve with weighted dual graph

[figure: −2; −1; −1; 0; −2; −2; −2; −2; −3; (α); (β)]

(Dual graphs can obviously be defined, more generally, when all singularities are ordinary double points.) Because of symmetry of the graph one expects that the cutting sets (α) and (β) lead to isomorphic towers. This is indeed the case because one can show that the situation carries an involution which interchanges (α) and (β). The tower of the last graph contains surfaces which are not contained in the Ramanujam tower, e. g. cut the cycle with multiplicities (1, 1). The resulting tree is not minimal and can be contracted to

[figure: −2; −1; 3; −3]

It belongs to a surface with \overline{κ} = −∞. By contraction of the part with negative weights we see that the surface is isomorphic to the complement of a cuspidal cubic in IP^{2}.

(3.16) Four Lines in general position. Let L_{1}, L_{2}, L_{3}, L_{4} ⊂ IP^{2} be four lines in general position. Then D = L_{1} ∪ L_{2} ∪ L_{3} ∪ L_{4} is already a normal crossing curve. The dual graph Γ(D) is the 1–skeleton of a 3–simplex. Up to symmetry there are two cutting sets Φ and Ψ, as indicated in the next figure.

[figure: c; a; b; Φ = {a, b, c}; d; a; b; Ψ = {a, b, d}]

The discriminant in case Ψ is computed in tom Dieck – Petrie [1989], (2.10).

We illustrate the computation in terms of the present algorithm. Since ⟨L_{i}⟩ is a generator of H^{2}(IP^{2}) a relation basis is

(3.17) R_{1} = L_{1} − L_{2}, R_{2} = L_{1} − L_{3}, R_{3} = L_{1} − L_{4}.

The indexing of the lines is chosen such that with respect to the figures above

(3.18) \begin{matrix}∂a = \{L_{1}, L_{4}\} & ∂b = \{L_{2}, L_{4}\} \\ ∂c = \{L_{1}, L_{2}\} & ∂d = \{L_{1}, L_{3}\}. \end{matrix}

We use the following notation for the multiplicities: m(L_{j}, E_{0}(x)) = x_{j}. The values (3.14) are then by (3.17)

λ(1, x) = x_{1} − x_{2}
λ(2, x) = x_{1} − x_{3}
λ(3, x) = x_{1} − x_{4}.

In case Φ only a_{1}, a_{4}, b_{2}, b_{4}, c_{1}, c_{2} are non-zero. Therefore in this case Φ

(3.19) \begin{matrix}Λ & = & | \begin{matrix}a_{1} & −b_{2} & c_{1} − c_{2} \\ a_{1} & 0 & c_{1} \\ a_{1} − a_{4} & −b_{4} & c_{1} \end{matrix} | \\ & = & b_{2}c_{1}a_{4} + b_{4}a_{1}c_{2}. \end{matrix}

Since the multiplicities are positive integers we see that the discriminant has never the value one. The tower does not contain homology planes. The simplest element in this tower arises when all multiplicities are 1. This amounts to blowing up the points a, b, c once. The resulting tree has 4 vertices, a central one with weight 1 and three edges attached to it with terminal weights −1. By contracting the three (−1)–curves one arrives at a single curve with weight 4 in IP^{2}. This must be a quadric.

A similar computation in case Ψ yields the discriminant

(3.20) Λ = (b_{2} − b_{4})d_{3}a_{1} + (d_{1} − d_{3})b_{2}a_{4}.

This tower contains infinitely many homology planes. They are actually contractible: The next Proposition shows that they are simply-connected, and a simply-connected acyclic manifold is contractible by the Whitehead-theorem of algebraic topology.

(3.21) Proposition. Surfaces in the towers of the Four Lines arrangement have abelian fundamental group.

Proof. The fundamental group of any surface V in the tower is a quotient of the fundamental group of IP^{2} \ ⋃ L_{j} = W, since W is obtained from V by removing a submanifold of real codimension two. It is well known that the fundamental group of W is abelian. □

In many interesting cases the selection function must have constant value one. In this case we can state:

(3.22) Proposition. If α(Γ_{0}D) is surjective, then α(Γ_{0}(d)) is surjective.

Proof. The proof will also show how to obtain a relation basis for α(Γ_{0}D(d)) from a relation basis for α(Γ_{0}D). We have

D(d) = C = π^{−1}(D) = D′ ∪ {E | E ∈ E_{π}}.

We construct a commutative diagram with horizontal isomorphisms

H^{2}(Y) ⊕ ZZ(E_{π}) ⟨π^{∗}, α(E_{π})⟩_{\overline{}}H^{2}(X)

[figure: α(Γ_{0}D) ⊕ id; α(Γ_{0}C)]
ZZ(Γ_{0}D) ⊕ ZZ(E_{π})  ^{σ \overline{}}ZZ(Γ_{0}C).

¿From (3.4) we have π^{∗}⟨D_{j}⟩ = ⟨D′_{j}⟩ + α(E)[y_{j}) for suitable y_{j} ∈ ZZ(E_{π}). In order to make the diagram commutative we have to define σ by σ(D_{j}) = (D′_{j}, y_{j}) and σ(E) = E for E ∈ E_{π}. The map σ sends a relation basis to a relation basis. □

4. The weighted graph

A basic invariant of a homology plane or any other variety in a tower is the weighted dual graph. This section collects some facts about graphs and trees which are related to the tower algorithm:

1. The computation of the weighted dual tree from multiplicities and vice versa.

2. The computation of the discriminant from the weighted trees.

Weighted graphs and trees are convenient for practical computations. They can be used to illustrate the theory. Finally they are useful as a heuristic tool.

Many of the results of this section are known, but there seems to be no convenient reference for our purposes.

The weighted graphs of a standard expansion are obtained from the multiplicities by a continued fraction algorithm. Therefore we recall some facts about continued fractions.

The symbol [c_{1}, …, c_{r}] for a continued fraction is defined inductively by [c_{1}] = c_{1} and (4.1) [c_{1}, …, c_{r}] = c_{1} − [c_{2}, …, c_{r}]^{−1}.

If 1 ≤ q < p are coprime integers, there is a unique continued fraction expansion

(4.2) \frac{p}{q} = [c_{1}, …, c_{r}]

with integers c_{i} ≥ 2. The determinant of the (r, r)–matrix (a_{ij}) with a_{ii} = c_{i} and a_{ij} = −1 if |i − j| = 1 is denoted D(c_{1}, …, c_{r}). The intersection matrix of the weighted tree

[figure: −c_{1}; −c_{2}; −c_{r}]

is the negative of the matrix just considered. Expanding the determinant with respect to the first row yields for r ≥ 3

(4.3) D(c_{1}, …, c_{r}) = a_{1}D(c_{2}, …, c_{r}) − D(c_{3}, …, c_{r}).

We compare this with (4.1) and obtain by induction on r ≥ 2

(4.4) [c_{1}, …, c_{r}] = \frac{D(c_{1}, …, c_{r})}{D(c_{2}, …, c_{r})}.

The relation (4.3) can be used to prove by induction on r ≥ 3 the following identity

(4.5) D(c_{1}, …, c_{r−1})D(c_{2}, …, c_{r}) − D(c_{1}, …, c_{r})D(c_{2}, …, c_{r−1}) = 1.

This relation shows that the right hand side of (4.4) is a quotient of coprime integers. It also shows what happens, if continued fractions are turned around. Namely:

(4.6)  If p/q = [c_{1}, …, c_{r}], q · q^{∗} ≡ 1 mod p, and 1 ≤ q^{∗} < p, then p/q^{∗} = [c_{r}, …, c_{1}].

Our aim is to characterize standard expansions by weights and multiplicities. We use the following notation. Let M_{0}(A_{−}) and M_{0}(A_{+}) be a pair of coprime integers. We set

(4.7) \begin{matrix}\frac{M_{0}(A_{−}) + M_{0}(A_{+})}{M_{0}(A_{+})} = [a_{−r}, …, a_{−1}] \\ \frac{M_{0}(A_{−}) + M_{0}(A_{+})}{M_{0}(A_{−})} = [a_{s}, …, a_{1}]. \end{matrix}

Associated to these data and to two further integers a_{−}, a_{+} is the following weighted linear tree.

(4.8) [figure: a_{−} + a_{−r} − 1; a_{−(r−1)}; a_{−1}; 1; a_{1}; a_{s−1}; a_{+} + a_{s} − 1; A−^{′}; E_{−(r−1)}; E_{−1}; E_{0}; E_{1}; E_{s−1}; A+^{′}]

Suppose A_{−} and A_{+} are irreducible curves in a projective surface X and x ∈ A∩B is a regular point of transverse intersection of A and B (= an ordinary double point of A∪B). Although it is not necessary to assume at this point that A_{−} and A_{+} belong to a normal crossing curve, we use the graph notation and illustrate this situation by (4.9).

(4.9) [figure: a_{−}; x; a_{+}; A_{−}; A_{+}]

Here −a_{±} = A_{±} · A_{±} are the weights. We may think of (4.9) as being an edge in some larger weighted graph. An expansion p : Z → X with exceptional set {x} is called standard if it has the following properties (compare (1.4)).

(4.10)

(1) The dual graph of p^{−1}(x) is a linear tree.

(2) p^{−1}(x) contains a unique (−1)–curve E_{0}.

(3) If p^{−1}(x) ≠ E_{0}, then p^{−1}(x) has two terminal components. One of them intersects A_{−} transversely and is disjoint to A_{+} and the other one intersects A_{+} transversely and is disjoint to A_{−}.

We denote, in accordance with (4.8), the exceptional divisors of a standard expansion p by E_{j}, −r < j < s, and assume that E_{i} ∩ E_{j} ≠ ∅ if and only if |i − j| ≤ 1. Finally let A±^{′} denote the proper transform of A_{±} and assume that A−^{′} ∩ E_{−(r−1)} ≠ ∅ and A+^{′} ∩ E_{s−1} ≠ ∅. Forgetting about points in A ∩ B which differ from x, the intersection pattern of A_{−} ∪ p^{−1}(x) ∪ A_{+} is codified by a weighted dual graph of the shape (4.8), called the tree of A_{−} ∪ p^{−1}(x) ∪ A_{+}. That the weights are actually given by a continued fraction algorithm (4.7) is the partial content of the next Theorem. Nevertheless the yet unknown weights are denoted as in (4.8). Recall that in this section we write a_{j} = −E_{j} · E_{j} etc. in order to avoid minus signs in the figures (until further notice).

(4.11) Theorem. Let p : Z → X be a standard expansion with exceptional set {x} ⊂ A ∩ B and use the notation introduced above. Let M_{0}(A_{±}) denote the multiplicity of E_{0} in p^{∗}(A_{±}). Then the following holds.

(1) M_{0}(A_{+}) = D(a_{−(r−1)}, …, a_{−1})

M_{0}(A_{−}) = D(a_{s−1}, …, a_{1}).

If r = 1 or s = 1, then the corresponding multiplicities are 1.

(2) The multiplicities M_{0}(A_{−}) and M_{0}(A_{+}) are coprime integers and the tree of A_{−} ∪ p^{−1}(x) ∪ A_{+} is (4.8) with weights determined by (4.7).

Given the curves A_{±} and x, the standard expansion of x is uniquely determined by the pair (M_{0}(A_{−}), M_{0}(A_{+})) and any pair of coprime positive integers arises for some such standard expansion.

Proof. We already know from (3.2) that there exists a relation of the type

(4.12) p^{∗}(A_{±}) = A′_{±} + ∑_{j=−(r−1)}^{s−1} M_{j}(A_{±})E_{j}.

(We let E_{j} denote the cohomology class determined by this curve. One can also read (4.12) as an equality of Cartier divisors.) By induction over the number of σ–processes involved in p one shows:

(4.13) M_{s−1}(A_{−}) = 1, M_{−(r−1)}(A_{+}) = 1.

In order to simplify the exposition we assume r ≥ 2, s ≥ 2 and leave the cases r = 1 or s = 1 to the reader.

Denote the intersection pairing by ⟨−, −⟩. For any expansion p : Z → X the relation ⟨p^{∗}C, D⟩ = ⟨C, p_{∗}D⟩ holds for divisors C in X and D in Z (see Hartshorne [1977], V(3.2)). In cohomological terms, p_{∗} is the Hopf Umkehr homomorphism associated to p. This relation implies in our case

(4.14) ⟨p^{∗}(A_{+}), E_{j}⟩ = 0.

Let

α_{+} = ∑_{j=−(r−1)}^{−1} M_{j}(A_{+})E_{j}

be the negative part of p^{∗}(A_{+}). From (4.14) we conclude

(4.15) ⟨α_{+}, E_{−1}⟩ = −M_{0}(A_{+}), ⟨α_{+}, E_{j}⟩ = 0 for − r < j < 2.

This can be viewed as a linear system for the unknown M_{j}(A_{+}), −r < j < 0. With the help of Cramer’s rule we compute

(4.16) M_{0}(A_{+}) = M_{−(r−1)}(A_{+})D(a_{−(r−1)}, …, a_{−1}).

Thus (4.13) and (4.16) yield the first assertion of (4.11.1).

We intend to show (4.11.2) by induction on the number of σ–processes in p. We leave it to the reader to check that the induction starts correctly. Given the expansion p, a further σ–process has to blow up either E_{0} ∩ E_{1} or E_{0} ∩ E_{−1}. This follows from the fact that we only allow standard expansions. Let us treat the case that E_{0} ∩ E_{1} is blown up. Denote the new multiplicities by the letter N. Then we have

(4.17) N_{0}(A_{±}) = M_{0})A_{±}) + M_{1}(A_{±}).

For the induction step we have to show

(4.18) \begin{matrix}\frac{N_{0}(A_{+}) + N_{0}(A_{−})}{N_{0}(A_{−})} = [a_{s}, …, a_{2}, a_{1} + 1] \\ \frac{N_{0}(A_{+}) + N_{0}(A_{−})}{N_{0}(A_{+})} = [a_{−r}, …, a_{−1}, 2]. \end{matrix}

We use the identity

(4.19) M_{0}(A_{+})M_{1}(A_{−}) − M_{0}(A_{−})M_{1}(A_{+}) = −1.

This identity is shown by induction on the number of σ–processes by using (4.17). Instead of verifying the first identity in (4.18) we use (4.6) and verify (with N_{±} = N_{0}(A_{±}) etc.)

(4.20) \frac{N_{+} + N_{−}}{N_{−}^{∗}} = 1 + \frac{M_{+} + M_{−}}{M_{−}^{∗}} .

¿From (4.17) and (4.19) we calculate

(M_{+} + M_{−})N_{−} − (N_{+} + N_{−})M_{−} = −1

and this implies easily

(4.21) M_{−}^{∗} = M_{1+} + M_{1−} = N_{−}^{∗}.

(4.17) and (4.21) yield (4.20). The second identity in (4.18) requires the verification of

\frac{N_{+} + N_{−}}{N_{+}^{∗}} = 2 − \frac{M_{+}^{∗}}{M_{+} + M_{1}} ;

it uses (M_{+} + M_{−})N_{+} − (N_{+} + N_{−})M_{+} = 1 and N_{+}^{∗} = M_{+} + M_{−}, M_{+}^{∗} = M_{+} − M_{1+} + M_{−} − M_{1−}. This finishes the inductive proof of (4.11.2).

Once we know the multiplicities M_{0}(A_{±}) of the expansion p we know by (4.11.2) the weights. The weights, on the other hand, determine uniquely the inverse contraction, because p, being a standard expansion, has a unique (−1)– curve at every step in the iteration of σ–processses (proof by induction). Hence p is uniquely determined by the weights.

Finally, we show by induction that any pair of coprime positive integers arises as the pair of multiplicities for some standard expansion. Given such integers M_{0}(A_{−}) and M_{0}(A_{+}) we can formally write down the tree (4.8). By properties of continued fractions, either a_{−1} = 2, a_{1} > 2 or a_{−1} > 2, a_{1} = 2. Suppose a_{1} = 2. Then we can formally contract the (−1)–curve arrive at a tree with weights

(4.22) [figure: a_{−2}; a_{−1} − 1; 1; a_{2}; a_{3}]

These weights arise again from a continued fraction expansion. Namely, if

\frac{U + V}{U} = [2, b_{1}, …, b_{r}], \frac{U + V}{V} = [c_{1}, …, c_{s}],

then

\frac{U}{U − V} = [b_{1}, …, b_{r}], \frac{U}{V} = [c_{1} − 1, c_{2}, …, c_{s}].

Now turn the continued fractions around (4.6). By induction we know that (4.22) is realizable. This finishes the proof of (4.11). □

For completeness, we determine all multiplicities in terms of weights.

(4.23) Proposition. Under the hypothesis of (4.11) the multiplicities M_{j}(A_{±}) in (4.12) are given as follows:

\begin{matrix}(1) & M_{j}(A_{+}) = D(a_{s} − 1, a_{s−1}, …, a_{(j+1)}), & 0 < j < s. \\ (2) & M_{j}(A_{+}) = D(a_{−(r−1)}, …, a_{−(j+1)}), & −r < j ≤ 0. \\ (3) & M_{j}(A_{−}) = D(a_{−r} − 1, a_{−(r−1)}, …, a_{−j}), & −r < j < 0. \\ (4) & M_{j}(A_{−}) = D(a_{s−1}, …, a_{j+1}), & 0 ≤ j ≤ s. \end{matrix}

Proof. The proofs for (2) and (4) are similar to the proof in the case j = 0. Let us prove (1) by downward induction on j. We have

−a_{+} = ⟨A_{+}, A_{+}⟩ = ⟨p^{∗}A_{+}, A′_{+}⟩
= ⟨A′_{+}, A′_{+}⟩ + M_{s−1}(A_{+})⟨A_{+}, E_{s−1}⟩
= −a_{+} − a_{s} + 1 + M_{s−1}(A_{+}).

This starts the induction. The induction step follows from

0 = ⟨p^{∗}A_{+}, E_{j}⟩ = M_{j+1}(A_{+}) − M_{j}(A_{+})a_{j} + M_{j−1}(A_{+})

by using (4.3), with M_{j+1}(A_{+}) replaced by 1 in case j = s − 1. □

We mention the special case that one of the multiplicities M_{0}(A_{±}) equals 1. This case was left to the reader in (4.11) and (4.23). If M_{0}(A_{+}) = 1 and M_{0}(A_{−}) = n, then (4.8) has the form (4.24).

(4.24) [figure: a_{−} + n; 1; 2; 2; a_{+} + 1; A′; E_{0}; E_{1}; E_{n−1}; A+^{′}]

Our next task is the computation of the discriminant in terms of weights.

(4.25) Theorem. Let p : Z(p) → X, B(p) be an object in a tower (1.6), p ∈ B|(D, d, Φ). Denote by D(p) the determinant det(Γ_{0}B(p), w) of the weighted dual graph B(p) (see section 2) and by Δ the discriminant of the tower (see Theorem B). Then |D(p)| = |Δ(m_{p})|^{2}.

The proof of this Theorem is based on some general topological facts which we recall first.

(4.26) Lemma. Let B be a compact, connected, oriented 4–manifold with boundary S = ∂B. Suppose H̃_{∗}(B; Q) = 0. Then S is a Q–homology 3–sphere, H_{2}(S; ZZ) = 0 and

|H_{1}(S; ZZ)| = |kernel i : H_{1}(B; ZZ) → H_{1}(B, S; ZZ)|^{2}.

Proof. By Poincaré duality H_{2}(B, S; Q) ≅ H^{2}(B; Q) = 0. The exact homology sequence of (B, S) yields H_{1}(S; Q) = 0. Since H_{1}(S, ZZ) is a torsion group we obtain from duality and universal coefficients

H_{2}(S) ≅ H^{1}(S) ≅ Hom(H_{1}(S), ZZ) = 0.

For similar reasons we have a commutative diagram

[figure: H_{2}(B); ≅; H^{2}(B, S); ≅; Ext(H_{1}(B, S), ZZ)]

[figure: i^{∗}]
[figure: H_{2}(B, S); ≅; H^{2}(B); ≅; Ext(H_{1}(B), ZZ).]

By algebra, cokernel i^{∗} ≅ kernel i. The exact sequence

0 → cok i^{∗} → H_{1}(Σ; ZZ) → ker i → 0

yields the last assertion of (4.26). □

Proof of (4.25). We apply the preceding Lemma in the following situation. Let B be a normal crossing curve in the projective surface Z. Let U ⊂ Z be a closed tubular neighbourhood of B in Z, diffeomorphic to the manifold which is obtained by plumbing the normal disk bundles of the components of B. Then B = X \ U^{∘} is diffeomorphic to V and has a boundary ∂B = S. If V = Z \ B is a Q–homology plane in a rational projective surface Z, then the exact homology sequence of (Z, B) implies H_{1}(Z, B) = 0, because H_{1}(Z) = 0 and B is connected. Therefore (4.26) yields in this case

|H_{1}(S, ZZ)| = |H_{1}(V; ZZ)|^{2}.

We know already that |H_{1}(V, ZZ)| = |Δ(m_{p})| for V = Z(p) \ B(p). It remains to be shown that |H_{1}(S, ZZ)| = |D(p)|. This uses the fact that S = ∂U. Since H_{2}(S) = 0, H_{1}(U) = 0 we have an exact sequence

0  \overline{}H_{2}(U) ρ_{\overline{}}H_{2}(U, S)  \overline{}H_{1}(S)  \overline{}0

and by duality and universal coefficients, H_{2}(U, S) ≅ H^{2}(U) ≅ Hom(H_{2}(U), ZZ), the map ρ is transformed into the adjoint H_{2}(U) → Hom(H_{2}(U), ZZ) of the intersection pairing H_{2}(U) × H_{2}(U) → ZZ. By the very construction of U this intersection pairing is specified by the intersection matrix of Γ_{0}B. Hence |H_{1}(S)| = |cok ρ| is the absolute value of the determinant of the intersection matrix. □

(4.27) The Determinant Algorithm. We use (4.25) to derive an algorithm which yields a computation of the discriminant as a multilinear function from the data (D, d, Φ) which specify a tower. Suppose x is an edge in the weighted dual graph of D(d) which belongs to x ∈ Φ.

(4.28) [figure: a_{−}; A_{−}; x; a_{+}; A_{+}]

(The dots illustrate the fact that x is part of a larger graph.) We have to “cut the cycle at x”. The first step in the algorithm replaces the p and (4.28) in D(d) formally by

(4.29) [figure: α_{−}; A_{−}; α_{+}; A_{+}]

The weights α_{±} are now formal expressions

α_{±} = a_{±} + 1 − \frac{M_{0}(A_{+}) + M_{0}(A_{−})}{M_{0}(A_{±})}

in indeterminates M_{0}(A_{±}), called multiplicity variables. Having applied this process successively to all edges x ∈ Φ we obtain a tree where some of the weights are no longer integers but rational functions in the multiplicity variables. Nevertheless, we still have the determinant of the weighted tree. Multiply the resulting determinant by the product of all multiplicity variables.

(4.30) Proposition. The rational function, resulting from the preceding algorithm, is (up to sign) the square of the discriminant Δ(D, d, Φ).

Proof. The proof is a generalization of the continued fraction algorithm (4.4) for the computation of the determinant of a linear tree. Namely, consider the determinants of the weighted trees T and T′ which are related in the following maner: There is an edge in T with weights b and c at its boundary vertices. The vertex with weight b is a terminal vertex of T. Remove this edge and replace the weight c by c − b^{−1} and call the resulting weighted tree T′. Then

(4.31) det(T) = b det(T′).

The proof is the same as for (4.3). The tree of B(p) is obtained by replacing an edge (4.9) by (4.8) and then removing the vertex E_{0} and the adjacent edges. Now use (4.31) and induction on the length r, s of the continued fractions. □

(4.32) Example. We apply (4.27), (4.30) to case Ψ in the Four Lines arrangement (3.16). We obtain a weighted tree of the following type.

[figure: (4); (3); (u, v); (a, b); (1); (2); (x, y); \begin{aligned}(1) & = & 3 − \frac{a + b}{a} − \frac{u + v}{v} \\ (2) & = & 2 − \frac{x + y}{x} \\ (3) & = & 2 − \frac{a + b}{b} \\ (4) & = & 3 − \frac{u + v}{u} + \frac{x + y}{y} \end{aligned}]

((1)-(4) show the actual weights and not their negatives. The labeled dotted lines indicate the “cuts” and their multiplicity variables.) The reader may wish to compute the determinant and check the result agains (3.20).

(4.33) Remark. The algorithm (4.27) can be applied to each connected weighted graph and leads to a rather curious combinatorial problem. Cutting cycles always yields a quadratic form in the multiplicity variables. In the geometric case this form is the square of a linear form. Simple examples show that this is not always the case. Which graphs have the property, that all possible cycle cuttings yield squares of linear forms?

Proof of the Addendum to Theorem A. If V (p) = Z(p) \ B(p) is a homology plane, then by (3.7.1) the map α(Γ_{0}B(p)) is surjective. There is a commutative diagram

ZZ(Γ_{0}B(p)) α(Γ_{0}(B(p))_{\overline{}}H^{2}(Z(p))

[figure: q; p_{∗}]
ZZ(Γ_{0}D(d)) ^{\overline{}α(Γ_{0}D(d))} H^{2}(X)

where q maps a component L of B(p) to p_{∗}(L); this is either a component of D(d) or zero. Since p_{∗} is surjective too, so is α(Γ_{0}D(d)). □

5. Birational morphisms

A fixed homology plane V can be contained in towers arising from different pairs (Y, D), D a curve in a minimal rational surface. We want to study this ambiguity more closely. We call D a plane divisor of V in case Y = IP^{2} and otherwise a minimal divisor of V. Plane divisors which are arrangements of lines are classified in tom Dieck [1990]. Suppose the surfaces X_{1} \ C_{1} and X_{2} \ C_{2} are isomorphic. An isomorphism is a birational morphism φ : X_{1} → X_{2} which can be written as α_{2}α_{1}^{−1} with expansions α_{i} : Z → X_{i} having their exceptional sets Σ(α_{i}) in C_{i}. Then C = α_{i}^{−1}(C_{i}) and α_{i} induces an isomorphism Z \ C ≅ X_{i} \ C_{i}.

Let φ : Y_{1} α_{1}_{\overline{}}Z α_{2}_{\overline{}}Y_{2} be a birational isomorphism and let D_{1} ⊂ Y_{1} be a curve such that Σ(α_{1}) ⊂ D_{1}. Let p_{1} : X_{1} → Y_{1} be a contraction, C_{1} ⊂ X_{1} a curve and D_{1} = p_{1}(C_{1}). Then there exists a commutative diagram of expansions

X_{2}  f_{\overline{}}X_{1}

[figure: p; p_{1}]
[figure: Y_{2}; α_{2}; Z; α_{1}; Y_{1}.]

Let C_{2} = f^{−1}(C_{1}), so that f induces an isomorphism X_{2} \ C_{2} ≅ X_{1} \ C_{1} and set D_{2} = p_{2}(C_{2}). Then D_{2} ⊂ α_{2}α_{1}^{−1}(D_{1}), but equality does not hold in general. In particular, if Y_{1} and Y_{2} are minimal rational surfaces and X_{1} \ C_{1} is a homology plane belonging to some tower of (Y_{1}, D_{1}), then the same homology plane belongs to a tower of (Y_{2}, D_{2}). In order to apply this in practice one has to understand the way in which D_{2} differs from α_{2}α_{1}^{−1}(D_{1}).

It suffices to study the case φ = α_{1} or φ = α_{2} and α_{i} is a single σ–process. The case φ = α_{2} is trivial, because then D_{2} = α_{2}(D_{1}).

So let α_{1} blow up the point x ∈ D_{1}. There are two cases: x ∈ Σ(p_{1}) or x ∉ Σ(p_{1}).

If x ∉ Σ(p_{1}) we have a diagram

X_{2}  ^{f \overline{}}X_{1}

[figure: p; p_{1}]
Z  ^{α_{1} \overline{}}Y_{1}

and f blows up the “same” point as α_{1} does. In this case pf^{−1}(C_{1}) = α_{1}^{−1}(D_{1}). If x ∈ Σ(p_{1}) we have a factorization p_{1} = α_{1}p

X_{2}  ^{= \overline{}}X_{1}

[figure: p; p_{1}]
Z  ^{α_{1} \overline{}}Y_{1}

and the exceptional curve E = α_{1}^{−1}(x) may or may not be contained in p(C_{1}), so that D_{2} is either the total or the proper transform of D_{1}. Therefore this case depends on the properties of (p, C_{1}).

As a special case we consider a quadratic transformation

φ(S) = α_{2}(S)α_{1}(S)^{−1} : IP^{2} → IP^{2}.

It is specified by 3 points S = {x_{1}, x_{2}, x_{3}} which are not collinear: The points are blown up, α_{1}(S) : Z → IP^{2}, and the proper transforms of their connecting lines are contracted, α_{2}(S) : Z → IP^{2}, see Hartshorne [1977], V(4.2.3). If φ(S) is a birational map of this type, then the components of D_{2} are given as follows:

(5.1) (1) The images of those components of D_{1} which are different from the connecting lines of S. (2) The images of those exceptional curves E(x_{i}) for which x_{i} ∈ Σ(p_{1}) and which are in the image of p : X_{2} → Z.

Thus D_{2} may have more or fewer components than D_{1}.

(5.2) Five Lines. Take four lines L_{i} ⊂ IP^{2} in general position. Add a further line L through two double points P, Q of ⋃ L_{i}. Blow up the points P and Q and contract the proper transform of L. Then one arrives at a configuration of 6 lines in IP^{1} × IP^{1}, up to isomorphism IP^{1} × {0, 1, ∞} ∪ {0, 1, ∞} × IP^{1}. This configuration has to be used if the proper transform of the exceptional curves E(P) and E(Q) are contained in the compactification divisor of the homology plane having L ∪ L_{1} ∪ … ∪ L_{4} as plane divisor. If this is not the case, say E(P) is missing, apply the quadratic transformation with S = {P, Q, E}, where R is a further double point. The resulting curve has the form M_{1} ∪M_{2} ∪M_{3} ∪M_{4}, where M_{i} ⊂ IP^{2} are lines, and M_{1}, M_{2}, M_{3} have a common point. This arrangement has been used (essentially) by Gurjar – Miyanishi [1987] to construct homology planes with κ̄ = 1.

Homology planes which are in towers of Four Lines (3.14) are contained in towers of Five Lines. In order to see this apply again a quadratic transformation with S = {P, Q, R}. But this time assume that the curve E(R) is not contained in the compactification divisor. Then, by applying (4.1), we see that Four Lines result.

(5.3) Contractible surfaces with κ̄ = 1. In tom Dieck – Petrie [1990] a simple construction of contractible homology planes with κ̄ = 1 was given. We showed indirectly that our surfaces were isomorphic to those of Gurjar – Miyanishi [1987]. Here is a direct proof by birational morphisms. Let C(n, b) ⊂ IP^{2} denote the curve given by x^{n} = y^{b}z^{n−b}, (n, b) coprime positive integers. Let L_{z} be the line z = 0. Blow up a regular point of C(n, b) and let V (n, b) denote the complement of the proper transform C′(n, b) ∪ L′_{z}. This was the way we defined the homology planes V (n, b). Now add the lines L_{x}, x = 0, and L_{y}, y = 0 and consider D = C(n, b) ∪ L_{x} ∪ L_{y} ∪ L_{z}. Resolve singularities and get a normal crossing curve with weighted dual graph (see tom Dieck – Petrie [1990], (2.7) for the computation)

[figure: −w(s); −w(1); −1; −w(−1); −w(−r + 1); −w(1); −w(−r); −w(−1); −1; −w(1); −w(s) + 1; L_{x}; −1; 0; −1; L_{z}]

using the continued fraction expansions

n/b = [w(s), …, w(1)], n/a = [w(−r), …, w(−1)]

with a = n − b, 2a > n. It is now easy to see that this graph can be contracted to the Gurjar-Miyanishi configuration of four lines: Begin the contraction with the left and right most (−1)–curves.

(5.4) An arrangement of Miyanishi-Sugie [1988]. They have considered the following curve D_{1} ∪ D_{2} ⊂ IP^{2}: The curve D_{1} has degree n and a single singularity (cusp) of multiplicity n − 1; the curve D_{2} is a line which intersects D_{1} in two regular points P, Q with multiplicities D_{1} · D_{2} = (n − m)P + mQ. They show that such arrangements lead to homology planes and, for m = 1, to contractible surfaces. We demonstrate that their surfaces arise from Five Lines (4.2). This shows, incidentally, that they are contractible.

Resolving the singularities of the Miyanishi–Sugie curve gives the following graph (in brackets: the number of (−2)–curves in a string).

[figure: −2; −2; −2; −2; −1; −1; −n; −2; −2; −n; 0; E; C; D; B; A; −1; (m−1); (n−m−1); (n−1)]
Add (formally) connection of type  ^{^{}−1}  between AE, BC and BD.

Then the resulting graph can be (formally) contracted to the graph of the six lines in IP^{1} × IP^{1} (4.2). Now reverse the procedure to show existence and our claim simultaneously. The three (−1)–curves which are used to cut the cycles correspond after contraction to three lines which have to be added to the Miyanishi-Sugie curve. We leave it as a small exercise for the reader to figure out the position of these lines.

We finally mention without proof:

(5.5) The Ramanujam tower. The tower (3.13) is contained in towers arising from a quadric with three tangents. The Ramanujam surface itself is contained in a tower of Five Lines.

References

Barth, W., C. Peters and A. Van de Ven: Compact Complex Surfaces. Berlin - Heidelberg - New York, Springer 1984.

tom Dieck, T.: Linear plane divisors of homology planes. J. Fac. Sci. Univ. Tokyo, Sec.IA 37 (1990), 33–69.

tom Dieck, T., and T. Petrie: Homology planes. An announcement and survey. Topological Methods in Algebraic Transformation Groups. Conf. Proc., Rutgers 1988. Birkhäuser (1989), 27–48.

tom Dieck, T., and T. Petrie: Contractible affine surfaces of Kodaira dimension one. Japanese J. of Math. 1990, to appear.

Fujita, T.: On the topology of non-complete algebraic surfaces. J. Fac. Sci. Univ. Tokyo, Sect.IA 29 (1982), 503–566.

Gurjar, R.V., and M. Miyanishi: Affine surfaces with κ̄ ≤ 1. Algebraic geometry and commutative algebra in honor of Masayoshi Nagata (1987), 99–124.

Gurjar, R.V., and A.R. Shastri: On the rationality of complex homology 2– cells, I and II. J. Math. Soc. Japan 41 (1989), 37–56, 175–212.

Hartshorne, R.: Algebraic Geometry. New York - Heidelberg - Berlin, Springer 1977.

Miyanishi, M., and S. Tsunoda: Absence of the affine lines on the homology planes of general type. Preprint 1990, Osaka University.

Ramanujam, C.P.: A topological characterization of the affine plane as an algebraic variety. Ann. of Math. 94 (1971), 69–88.

Zaĭdenberg, M.G.: Isotrivial families of curves on affine surfaces and characterization of the affine plane. Math. USSR Izvestiya 30 (1988), 503–532. (Corrections to appear.)