Optimal rational curves and homology planes

Tammo tom Dieck

Version of November 18, 2000

I Introduction

This paper is concerned with the construction of acyclic affine surfaces from plane algebraic curves. Foundational material for this construction can be found in tom Dieck – Petrie [1989,1991]. Here we concentrate on the curves and explicit examples.

Surfaces will always be connected, 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}(V; 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. 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. By the basic rationality theorem of Gurjar – Shastri [1989] we can and will assume that X is a rational projective surface and C a rational curve.

Suppose X \ C = V is a homology plane and p : X → Y a contraction to a minimal rational surface Y. Then D = p(C) will be called a minimal divisor of V. In case Y = IP^{2} is the projective plane, we call D a plane divisor of V.

Only very special rational curves can be minimal divisors of Q–homology planes. In qualitative terms: If D = D_{1} ∪ … ∪ D_{n} is the decomposition into irreducible components, then the D_{i} should have very few intersection points or singularities with several branches. For instance, if D ⊂ IP^{2} is irreducible, then D must be a cuspidal curve; this means, D is a topologically embedded 2–sphere IP^{1}.

The minimal divisors of a homology plane V are by no means uniquely determined. In fact, by applying an arbitrary Cremona transformation to a plane divisor of some V we obtain again a plane divisor of V. Therefore we are looking for some kind of normal form. The normal forms should be chosen so as to meet certain criteria: They should be as simple as possible (fewest number of components, lowest degree of components). They should be maximal in the following sense: It is shown in tom Dieck – Petrie [1991] that a minimal divisor of a homology plane gives raise to infinite families of surfaces, called towers of surfaces. (The members of a tower are in general Q–homology planes.) The maximality condition askes for curves for which the corresponding towers are not contained in larger towers of other curves. For instance, the tower of a cuspidal curve D ⊂ IP^{2} consists of the complement IP^{2} \ D alone. But all known cuspidal curves are members of infinite towers. We offer two conjectures which help to explain these heuristic conditions.

(0.1) Conjecture. Every homology plane has a plane divisor which consists of lines and quadrics.

Unfortunately, it is very difficult to classify arrangements of lines qnd quadrics which can be plane divisors of homology planes. (For lines see tom Dieck [1990]; for lines and quadrics Neusel [1992].) An affirmative answer to the corresponding conjecture for (rational) Q–homology planes would imply that a cuspidal curve can be transformed into a line by a Cremona transformation (compare Matsuoka – Sakai [1989] for remarks concerning the latter problem). Actually, I believe that the methods of this paper lead to a construction of all cuspidal curves.

Let D = D_{1} ∪ … ∪ D_{n} ⊂ Y be a rational curve in a minimal rational surface Y. There is a minimal expansion p : X → Y such that C = p^{−1}(D) has normal crossings (embedded resolution of singularities). The intersection pattern of C is usually codified in its weighted dual graph ΓC. The curve D is called optimal if ΓC is connected and has n − b_{2}(X) cycles. Here b_{2} is the second Betti number; the number of cycles of a connected graph Γ is defined to be 1 − χ(Γ), with χ(Γ) the Euler characteristic of (the geometric realization of) Γ. The term “optimal” refers to the algorithm for the construction of homology planes from minimal divisors which was described in tom Dieck – Petrie [1991]. This algorithm requires the choice of a selection function and the choice of a cutting set. For optimal curves there is only a trivial selection function with constant value one.

(0.2) Conjecture. Homology planes have optimal minimal divisors.

Whereas (1.1) may well be true for Q–homology planes, (1.2) seems to require integral homology planes. Also it is not enough to deal only with plane divisors. Optimal plane divisors will not, in general, consist of lines and quadrics.

One purpose of this paper is to describe the (maximal) optimal rational curves which have been found so far (up to Cremona transformations). Since homology planes of logarithmic Kodaira dimension κ̄ ≤ 1 have been analyzed successfully by Gurjar – Miyanishi [1987], we always concentrate on surfaces of general type κ̄ = 2. There probably exist no more than those coming from the curves described here. This belief is the main justification for this paper. But, loosely speaking, I conjecture that there exists a finite number of towers which comprise all homology planes of general type.

The results of this paper suggest a number of more technical conjectures which I will explain in due course.

In section 2 we construct the optimal curves and show that they are uniquely determined by their weighted dual graph. In section 3 we present some numerical results for the construction of homology planes. In section 4 we verify that the known homology planes have very special logarithmic Chern numbers.

II Optimal curves

The curves are described by their intersection pattern (weighted dual graph, singularities). Existence and uniqueness up to automorphism of the minimal rational surface is shown. Recall that the minimal rational surfaces are IP^{2} and the Hirzebruch surfaces Σ(n), n = 0, 2, 3, . . .. We also show how our optimal curves are related to arrangements of lines and quadrics in order to check (1.1).

A. Six lines in Σ(0).

Note that Σ(0) = IP^{1} × IP^{1}. The arrangement consists of

L_{i} = IP^{1} × i,  M_{i} = i × IP^{1},  i = 0, 1, ∞ ∈ IP^{1}.

An equivalent optimal arrangement of Five Lines in IP^{2} is the following: Take four lines in general position L_{1}, …, L_{4} and add a further line L which passes through two double points x and y of L_{1} ∪ … ∪ L_{4}. In order to obtain a normal crossing curve one has to blow up x and y. Then the proper transform of L becomes a (−1)–curve. Contracting this (−1)–curve leads to an arrangement of Six Lines in Σ(0) as described above. The weighted dual graph has 4 cycles and is (the non-planar graph) given in Figure A. This is the only arrangement with 4 cycles.

[figure: 0; 0; 0; 0; 0; 0]

Figure A

We remark that Four Lines in general position in IP^{2} are an optimal curve (three cycles). But the towers of this arrangement are contained in towers of the arrangement (A) above. In order to see this, ones has to cut one of the three cycles with multiplicities (1, 1) (see tom Dieck – Petrie [1991] for terminology and background). Cutting a cycle with these multiplicities removes an edge in Figure A and replaces the weights of the adjacent vertices by −1. Contracting these (−1)–curves results in the graph of Four Lines (see tom Dieck – Petrie [1991], (5.2)).

B. Quadric with three tangents.

Up to automorphisms of IP^{2} there exists a unique curve consisting of a regular quadric Q and three of its tangents T_{1}, T_{2}, T_{3}. The symmetric group S_{3} still acts on any such arrangement by projective automorphisms. Blowing up the points of tangency (twice) produces a normal crossing curve. The weighted dual graph is given in Figure B.

As a matter of notation we display the negatives of the actual weights in order to avoid minus-signs.

[figure: 1; 1; 1; 1; 2; 2; 1; 1; 2; 2]

Figure B

C. Quadric, two tangents, line through double point.

Let Q be a quadric with two tangents T_{1}, T_{2} and a further line L through T_{1} ∩T_{2}. Any such arrangement has a projective symmetry group ZZ/2 × ZZ/2: Reflection in L, interchanging T_{1} and T_{2}; reflection in the line through Q∩T_{i}. The weighted dual graph of a minimal resolution is given in Figure C.

[figure: 2; 1; 0; 1; 2; 0; 2; 1; 2]

Figure C

(0.3) Proposition. The surfaces in towers of B and C are obtainable from the following arrangement of Seven Lines (Figure 1): Let L_{1}, …, L_{4} be four lines in general position. Add the three possible lines M_{1}, M_{2}, M_{3} which connect the double points of L_{1} ∪ … ∪ L_{4}.

[figure: □; □; □]

Figure 1

Proof. Note that the symmetric group S_{4} acts on this arrangement by permuting the L_{j}. The construction of towers requires the choice of a selection function (see tom Deick – Petrie [1991]). In the case of an arrangement of lines a selection function d assigns a value 0 or 1 to a point of multiplicity greater than two. The combinatorics requires in our case that d assumes the value 0 on three non-collinear points. (Value 0 means: When this point is blown up the corresponding exceptional divisor will not be included in the compactification divisor of the homology plane. If three collinear points have value zero, then the resulting graph is no longer connected.) Up to symmetry these are the cases marked □ and ◯ in Figure 1. Case ◯ leads to the graph B and case □ to the graph C.

In the case of graph B we can contract successively (−1)–curves. We end up with four curves Q, T_{1}, T_{2}, T_{3} with self-intersections (respectively) 4, 1, 1, 1 and intersection numbers |Q · T_{j}| = 2. All this is read off from the graph. Thus any such graph arises from an arrangement B. A similar argument applies to case C. □

As a matter of notation we write S · T = ∑ n_{i}x_{i} when the irreducible curves S and T intersect in the points x_{i} with multiplicity n_{i}. The intersection number |S · T | is then given by ∑ n_{i}.

D. Cubic with two lines.

Let C ⊂ IP^{2} be a cubic with a cusp s. There is a unique point p ∈ C which has a tangent L of order 3, i.e. C · L = 3p. Let M be an ordinary tangent to C in a regular point r. It intersects M in a further regular point q. The points q, r are different from p, since C \ s carries a group structure (isomorphic to the additive group C) such that the intersections of C \ s with a line add up to zero in this group structure. A cubic with a cusp is projectively unique and still carries a C^{∗}–action with fixed points p, s and orbit C \ {p, s}. This shows that the configuration C ∪ L ∪ M is projectively unique. The dual graph is specified in Figure D.

[figure: 2; 2; 1; 2; 1; 1; 2; 2; 1; 3; 2]

Figure D

E. Four sections in Σ(2).

Let π : Σ(2) → IP^{1} be the ruling , F a general fibre of π and E the section with self-intersection |E · E| = −2. The arrangement consists of curves Q_{2}, Q_{1}, L, E linearly equivalent to

Q_{2} ∼ 3F + E, Q_{1} ∼ 2F + E, L ∼ 2F + E.

The intersection pattern is given as follows

Q_{2} · E = x, Q_{2} · Q_{1} = 3y, Q_{1} · L = 2v, Q_{2} · L = 2z + u.

Here x, y, z, u, v denote 5 different points. From this intersection pattern one computes the dual graph of a minimal resolution to be as in Figure E.

Figure E (0.4) Proposition. An arrangement with the intersection pattern as above exists and is unique up to automorphism of Σ(2).

[figure: 2; 1; 1; 2; 2; 2; 1; 3; 2; 2; 1]

Proof. We start with the following arrangement of lines L, M and quadrics Q_{1}, Q_{2} in IP^{2}. The intersection pattern is specified as follows:

Q_{1} ·Q_{2} = a+3c, M ·Q_{1} = 2a, M ·Q_{2} = a+d, L·Q_{1} = 2b, L·Q_{2} = d+e, L·M = d.

The construction is done in the following order:

(1) Choose Q_{1}, Q_{2} with the given intersection pattern. This is projectively unique.

(2) M is the tangent to Q_{1} in a. Then M is not tangential to Q_{2} in a and produces the other intersection d with Q_{2}. We must have c ≠ d.

(3) From d draw the second tangent L to Q_{1}. This determines L · Q_{1} = 2b. We have b ≠ c since otherwise L would be tangent to Q_{2}. We find e from L · Q_{2} and e is different from b and c. Having chosen Q_{1}, Q_{2}, the construction is uniquely determined.

We apply the following expansions and contractions to the arrangement. (α) Blow up a. This yields a (−1)–curve E and the proper transforms

L′, M′, Q′_{1}, Q′_{2}.

(β) E, M′ and Q′_{1} have a common point. We blow up this point. Then M′ becomes a (−1)–curve M′′.

(γ) Contract M′′.

The proper transforms of L, E, Q_{1}, Q_{2} are the curves we are looking for.

Given an arrangement L, E, Q_{1}, Q_{2} as in the beginning we can certainly reverse (α)−(γ) and arrive at a uniquely determined arrangement of lines and quadrics.□

[figure: x; y] with ∞ − line

Figure 2 (0.5) Proposition. The surfaces in towers D and E are obtainable from the following arrangement of Nine Lines in IP^{2} (Figure 2): The lines

z_{1} = 0, 1, 2; z_{2} = 0, 1, 2; z_{1} = z_{2}; z_{1} = z_{2} + 1;

and the infinite line (affine coordinates z_{1}, z_{2}).

Proof. A selection function d must have value one on two points of multiplicity 3. In order to obtain a connected graph one of these points has to be a point of multiplicity 3 on the infinite line. By symmetry of the figure the other point must be x or y. The case d(x) = 1 leads to the graph D and the case d(y) = 1 leads to the graph E. By contracting successively (−1)–curves in such graphs one arrives at a system of curves with intersection pattern as described under D and E. We have already seen that the intersection pattern determines the arrangement involved uniquely. □

F. Four sections in Σ(2).

As in case (E) we find curves with linear equivalence classes

C ∼ 3F + E, L_{1} ∼ L_{2} ∼ 2F + E, E.

But this time the intersection pattern has the following structure:

C · E = x, C · L_{1} = 3y, C · L_{2} = 3z, L_{1} · L_{2} = u + v.

This intersection pattern leads to the following dual graph (Figure F).

[figure: 2; 2; 1; 1; 2; 2; 1; 1; 2; 2]

Figure F

(0.6) Proposition. An arrangement with the intersection pattern as above exists and is unique up to automorphism of Σ(2).

Proof. We start with the following arrangement in IP^{2}: Let C be a cubic with node c. Let T be one of the tangents to C in c. Then there is no further intersection of C and T. There are exactly 3 regular points of C with tangents of order 3 (flexes). Let L_{1} and L_{2} be two of these tangents in d and e. We obtain intersections T · L_{1} = a, T · L_{2} = b, L_{1} · L_{2} = c. Up to automorphism of IP^{2} there is exactly one such arrangement.

We now apply the following expansions and contractions to this arrangement: (α) Blow up the point c. Let E be the corresponding (−1)–curve. (β) Blow up the intersection of E with the proper transforms C′ and T′. (γ) Contract the resulting (−1)–curve T′′.

The proper transforms of C, L_{1}, L_{2} and E constitute the arrangement we are looking for. □

G. Nodal cubic with two tangents.

This is the subsystem C ∪ L_{1} ∪ L_{2} which was described in the proof of (2.4). It leads to the following dual graph (Figure G).

[figure: 2; 2; 1; 2; 2; 2; 1; 2; 1; 1]

Figure G

(0.7) Proposition. The surfaces in towers F and G are obtainable from the following arrangement of Nine Lines in IP^{2}:

z_{1} = 1, ω, ω^{2}; z_{2} = 1, ω, ω^{2}; z_{1} = z_{2}; ωz_{1} = z_{2};

and the infinite line (ω ≠ 1, ω^{3} = 1). (This arrangement is not realizable over the real numbers.)

Proof. A selection function must have value one on three points of multiplicity 3. The points can be x, y or x, z (up to symmetry) with x = {z_{1} = 1} ∩ {z_{2} = 1}, y = {z_{1} = 1} ∩ {z_{2} = ω}, z = {z_{1} = ω} ∩ {z_{2} = ω}. The proof is finished as for (2.3). □

H. Cubic, quadric, line.

(0.8) Proposition. There exists an arrangement in IP^{2} consisting of a cuspidal cubic C, a regular quadric Q, and a line L with the following intersection pattern:

C · L = 3z, C · Q = 3x + 2z + y, Q · L = 2z.

The points x, y, z ∈ C are regular.

Proof. There certainly exist C and L with C · L = 3z. We use again the group structure on the regular part of C. Six regular intersection points x_{j} of C and Q (counted with multiplicities) must satisfy x_{1} + … + x_{6} = 0 in this additive group structure. Since C · L = 3z, the point z corresponds to the zero element of this group. The points x and y therefore have to satisfy 3x + y = 0. The quadric which is determined by these conditions is regular: Otherwise there would exist a line intersecting C in a subset of {x, y, z} and this is impossible. Again the configuration is unique up to automorphism of IP^{2}. □

The dual graph of the minimal resolution is given in Figure H.

[figure: 2; 2; 1; 1; 2; 2; 2; 1; 3; 2; 1; 3]

Figure H

(0.9) Remark. The surfaces in towers of H are obtainable from the following arrangement: A quadric Q with three tangents T_{1}, T_{2}, T_{3} together with two lines L_{1} through T_{1} ∩ T_{2} and Q ∩ T_{3} = y and L_{2} through Q ∩ L_{1} \ {y} and T_{1} ∩ T_{3}.

In order to prove this one has to find a selection function which produces the graph H. Any configuration with this graph can be contracted to the arrangement of (2.6): Contract successively (−1)–curves which show up in the curves of the graph. This last remark applies to all other graphs.

I. Quartic with bitangent.

A quartic with three cusps S ⊂ IP^{2} is projectively unique (Steiner quartic) and has a bitangent T. The graph of the minimal resolution of S ∪T is given in Figure

[figure: 2; 1; 3; 1; 2; 1; 2; 2; 1; 3; 3; 3; 1; 2; 6]

Figure I

This arrangement was used in tom Dieck [1990] to produce homology planes with ZZ/3–action. There it is also shown that the towers of (I) can be obtained from an arrangement of two quadrics and three lines.

There is another arrangement which yields the same result. There exists a cuspidal quintic D with four cusps x_{1}, x_{2}, x_{3}, and y (compare Namba [1984], p. 182, curve number 6). The points x_{j} are ordinary cusps of multiplicity two. There exists a tangent L in y which intersects D in a single further point z which is a regular point of D. In order to see that this arrangment D ∪ L exists and yields the same result as S ∪ T above, one blows up one of the points S ∩ T. Then the graph of (I) is modified so as to contain the piece

[figure: 1; 2; 1; 3; 2; 2; 7]

and the vertical string can be contracted to the singular point y. This construction also shows a ZZ/3–symmetry on the quintic which permutes the cusps.

J. A 2–section and two sections in Σ(2).

There exists an arrangement S, L, E in Σ(2) with the following intersection properties

S ∼ 5F + 2E, L ∼ 2F + E, E
S · E = a, S · L = 4b + c.

The curve S has two cuspidal singularities and a, b, c are regular points of S. The intersection pattern yields the graph of Figure J.

[figure: 2; 1; 3; 2; 1; 3; 2; 2; 2; 1; 2; 1; 5; 2]

Figure J

(0.10) Proposition. An arrangement as above exists and is unique up to automorphism of Σ(2).

Proof. Start with two quadrics Q_{1}, Q_{2} in IP^{2} such that Q_{1} ·Q_{2} = 4x and choose a further point y ∈ Q_{1}. The triple (Q_{1}, Q_{2}, y) is unique up to automorphism of IP^{2}. Let T_{1}, T_{2} be the tangents of Q_{2} which pass through y. Let L be the line through Q ∩ T_{1} and Q_{1} ∩ T_{2} and let M denote the line through y and one of the intersections Q_{2} ∩ L. By symmetry it does not matter which of the points in Q_{2} ∩ L is chosen.

Now apply a plane Cremona transformation with centers y, Q_{1} ∩ L to Q_{1}, Q_{2}, T_{1}, T_{2}, L, M. Then Q_{2} becomes a quartic Q′_{2} with two cusps and a node z; and M and Q_{1} become lines M′, Q′_{1}. The next step is to contract the curves L, T_{1}, T_{2}. Moreover M′ is one of the tangents of Q′_{1} in y. Now one blows up y twice and contracts the proper transform of M′. The proper transforms S of Q_{2}, L of Q_{1}, and E of the exceptional divisor of y are the arrangement we are looking for. By reversing this procedure we obtain uniqueness, since the arrangement Q_{1}, Q_{2}, T_{2}, T_{2}, L, M was unique up to automorphism. □

K. Quartic with tangent.

In the previous section J we obtained a quartic S with two cusps, a node and a tangent L of order 4. The curve S ∪ L is optimal and has the graph:

[figure: 2; 1; 3; 2; 1; 3; 3; 1; 2; 2; 2; 1; 4]

Figure K

The corresponding tower can be obtained from the arrangement Q_{1}, Q_{2}, T_{1}, T_{2}, L of the previous section. There exists an involution on IP^{2} which interchanges the cusps of S and fixes S ∩ L and the node of S.

L. A quintic with cuspidal tangent.

(0.11) Proposition. There exists a quintic Q ∈ IP^{2} with three cusps. The multiplicity sequence of each cusp is (2, 2). The tangent T at a cusp intersects Q in a single further point which is regular. A quintic with these types of cusps is projectively unique. There exists a ZZ/3–action on IP^{2} which permutes the cusps of Q.

Proof. We give a geometric contruction which exhibits the ZZ/3–symmetry. The starting point is an arrangement of a quadric with six or seven lines.

Let S be a quadric with three tangents T_{1}, T_{2}, T_{3}. Set x_{ij} = T_{i} ∩ T_{j}. There exist lines L_{k} through x_{ij}, k ∉ {i, j} such that L_{k} ∩ L_{l} ∈ S for k ≠ l. Existence is shown as follows by working in the real plane: Let S be a circle about the origin in the plane. Suppose the T_{i} are permuted under a 2π/3–rotation about the origin. Add lines L_{k} such that L_{k} ∩L_{l} ∈ S and such that the L_{k} are permuted under a 2π/3–rotation. Now rotate the configuration of the L_{k} until L_{k} passes through X_{ij}, k ∉ {i, j}. Starting from this construction it is also possible to show uniqueness of the arrangement.

The quintic Q is now obtained by applying the following Cremona transformation: Blow up the points x_{ij} and L_{k} ∩L_{l}. Then T_{i} becomes a (−1)–curve and L_{i} a (−2)–curve. Contract these (−1)–curves and then the (−2)–curves. The proper transform of S is Q. It is clear that this process is ZZ/3–equivariant. Reverse this procedure to show uniqueness of Q.

If we add a line through x_{13} and S ∩ L_{2} in the primary configuration, then T is the proper transform of this line under the birational map of the previous paragraph. □

The curve Q ∪ T is optimal. The graph of the minimal resolution contains a (−1)–curve. We contract this curve and arrive at the graph of Figure L.

[figure: 2; 1; 3; 2; 2; 1; 1; 2; 2; 2; 3; 2; 4]

Figure L

(0.12) Remark. The graph of figure L can be contracted to a curve S′ ∪ T′ in IP^{2}. In this case S′ is again a quintic, but this time with two cusps and a tacnode t, and T is a line passing through t. By applying a Cremona transformation with centers x_{ij} to the primary configuration one obtains another interesting one: A Steiner quartic together with three lines through its cusps; the lines intersect on the quartic. The Cremona transformation with centers L_{k} ∩ L_{l} yields another interesting configuration of three quadrics with a common tangent.

M. Nodal cubic, quadric, line.

(0.13) Proposition. Let C ⊂ IP^{2} be a nodal cubic. There exists a quadric Q and a line L such that we have the following intersections:

C · Q = 4x + 2y, C · L = 3y, Q · L = 2y.

Proof. We use the group structure C^{∗} on the complement of the double point of C. Then we take y = exp(2πi/3) and x = exp(2πi/12). There exists a regular quadric Q which intersects C in x_{1}, …, x_{6} if and only if ∏ x_{i} = 1 in this group structure. Since x^{4}y^{2} = 1 we can realize the intersection pattern C · Q as stated. A similar argument applies to L. □

The dual weighted graph is displayed in Figure M.

[figure: 2; 2; 2; 2; 1; 2; 1; 2; 2; 2; 1]

Figure M

The towers of the resulting Q–homology planes can be obtained by the following arrangement of lines L_{1}, …, L_{5} and a quadric Q (see Neusel [1992]): L_{1}, L_{2}, L_{3} are tangents to Q and L_{4} passes through Q ∩ L_{1} and Q ∩ L_{2}. Finally L_{5} passes through L_{1} ∩ L_{2} and L_{3} ∩ L_{4}.

N. Four sections in Σ(0).

[figure: 2; 2; 1; 3; 2; 1; 2; 2; 1; 2; 1]

Figure N

(0.14) Proposition. There exist four sections S_{1}, …, S_{4} of pr_{1} : IP^{1} × IP^{1} → IP^{1} with the following properties: S_{1} = IP^{1} × {y_{i}}, i = 1, 2, 3 and S_{4} · S_{1} = 3x_{1}, S_{4} · S_{2} = 2y_{1} + y_{2}, S_{4} · S_{3} = 2z_{1} + z_{2}.

Proof. We start with configuration D and add the line which passes through the transvers intersection of C and M. We then blow up the cusp s and L∩M = t and contract the proper transform of the line which connects s and t. The proper transforms of L, M, and Q become S_{1}, S_{2}, and S_{3} and the exceptional divisor of s becomes S_{4}. □

The towers of Q–homology planes are obtainable from the following arrangement of a quadric Q and lines L_{1}, …, L_{6}: The lines L_{1}, L_{2}, L_{3} are tangents of Q, L_{4} passes through L_{1} ∩ L_{2} and Q ∩ L_{3} and L_{5} (L_{6}) passes through Q ∩ L_{4} and L_{1} ∩ L_{3} (L_{2} ∩ L_{3}).

(0.15) Remark. A surface V is said to admit a C^{n∗}–fibration if there is a morphism p : V → U onto a curve U such that the general fibre of p is the n–punctured affine line C. From the construction of our optimal curves it is easy to see, that their complement admits a C^{2∗}–fibration (cases A,B,C) or a C^{3∗}– fibration (the other cases). Since the cases A, B, C also admit C^{3∗}–fibrations we state:

(0.16) Conjecture. Every homology plane of general type admits a C^{3∗}– fibration.

The classification of C^{3∗}–fibrations will be the subject of another essay.

III Discriminants

We recall from tom Dieck – Petrie [1991] that the construction of towers from a minimal divisor requires:

(1) The choice of a selection function.

(2) The choice of a cutting set. Since we are dealing with optimal curves there is no choice of a selection function. A cutting set is a set of edges in the graph such that its omission produces a tree. The surfaces in a tower belonging to a given cutting set are parametrized by multiplicities: To each edge in a cutting set is assigned a pair (u, v) of positive coprime integers u and v, called multiplicities. There is a multilinear function in the multiplicities, called discriminant of the tower, such that its absolute value gives the order of the first homology group of the associated surface in the tower (provided this value is non zero). The surface is a homology plane if and only if the discriminant has absolute value one. The discriminant is calculated via the following algorithm:

Start with a curve C = C_{1} ∪ … ∪ C_{n} in a minimal rational surface Y and let C_{i} ∈ H^{2}(Y; ZZ) also denote the Poincaré dual of the cycle C_{i}. Let R_{1}, …, R_{k} denote a basis of the relations among the C_{i}; the R_{j} are certain integral linear combinations of the indeterminates C_{i}. Now blow up the surface to resolve the singularities of C. Rewrite R_{j} in terms of the proper transforms of the C_{i} and the exceptional divisors of the expansion. From the resulting linear combinations R′_{j} the discriminant is produced in the following manner:

Let x_{e} be an edge in the cutting set with vertices (= curves) A and B. Replace A and B in the R′_{j} by a(A)x_{e}, a(B)x_{e}, where a(A), a(B) ∈ ZZ are the multiplicities. If A occurs in several edges of the cutting set, replace A by the sum of the corresponding terms a(A)x_{e}. Set A = 0, if A is not a vertex of an edge in a cutting set. It is a fact that k is also the cardinality of a cutting set. The coefficients of the x_{e} in the R′_{j} after these replacements yield a (k, k)–matrix whose determinant is the value of the discriminant at the given multiplicities.

Since some of our graphs have a large number of cutting sets (e.g. B has 11, up to symmetry), we give only the linear polynomials R′_{j} in those variables which can appear in cutting sets. As for notation, we use the symbols for the curves which were introduced in the definition of the arrangement (A) to (N) in section 2. Moreover we need notation for some of the exceptional divisors: We draw again the relevant graphs and specify the symbol for the divisor at the corresponding vertex.

Arrangement A T. tom Dieck 17

[figure: M_{∞}; L_{∞}; M_{1}; L_{1}; L_{0}; M_{0}]  \begin{matrix}L_{0} − L_{1}, L_{0} − L_{∞} \\ M_{o} − M_{1}, M_{0} − M_{∞} \end{matrix}

Arrangement B

[figure: T_{1}; T_{2}; A_{1}; A_{2}; Q; T_{3}; A_{3}]  \begin{matrix}T_{1} − T_{2} + 2A_{1} − 2A_{2} \\ T_{2} − T_{3} + 2A_{2} − 2A_{3} \\ Q − 2T_{3} + 2A_{1} + 2A_{2} − 2A_{3} \end{matrix}

18  III. Discriminants  T. tom Dieck

Arrangement C

[figure: T_{1}; E; T_{2}; L; A_{1}; Q; A_{2}]  \begin{matrix}Q + 2A_{1} + 2A_{2} − 2L − 2E \\ T_{1} − T_{2} + 2A_{1} − 2A_{2} \\ L − T_{1} − 2A_{1} \end{matrix}

Arrangement D

[figure: E; C; L; M; F]  \begin{matrix}C − 3L + 2F − 6E \\ L − M + 3E − 2F \end{matrix}

Arrangement E

[figure: C; L; B; Q_{1}; Q_{2}; A]  \begin{matrix}Q_{1} − L + 3A − 2C \\ 2Q_{2} − 3L + 6A − 6B − 2C \end{matrix}

Arrangement F

[figure: A_{1}; L_{1}; A_{2}; L_{2}; C] \begin{matrix}L_{1} − L_{2} + 3A_{1} − 3A_{2} \\ 2C − 3L_{1} − 6A_{1} + 3A_{2} \end{matrix}

T. tom Dieck 19 Arrangement G

[figure: E; C; A_{1}; L_{1}; A_{2}; L_{2}]  \begin{matrix}L_{1} − L_{2} + 3A_{1} − 3A_{2} \\ C − 3L_{1} − 6A_{1} + 3A_{2} + 2E \end{matrix}

Arrangement H

[figure: A; C; B; Q; E]  \begin{matrix}C + 3E − 6A − 4B \\ Q + 3E − 4A − 2B \end{matrix}

Arrangement I

[figure: A_{1}; A_{2}; T; S]  S − 4T − 6A − 6B

Arrangement J

[figure: A; L; S]  2S − 5L − 8A

Arrangement L

[figure: E′; E; Q; T]  Q − 10E′ − 6E − 5T

For the convenience of the reader and by way of example we explain one of the arrangements in some detail. Consider case H, originating from a cubic C, a quadric Q, and a line L. A relation basis in H^{2}(IP^{2}; ZZ) is obviously C = 3L, Q = 2L. Now one has to resolve the singularities in x, z given by C · Q = 3x + 2z + y. This requires to blow up x three times and z twice. The images of the (Cartier) divisors C, Q, L are then (using e.g. Hartshorne [1977], V.3) respectively

C + 3A + 2B + B′ + 3E + 2E′ + E′′,
Q + 2A + 2B + B′ + 3E + 2E′ + E′′,
L + 3A + 2B + B′

where A, B, B′, E, E′, E′′ are names for exceptional divisors and L, B′, E′, E′′ are not involved in cycles of the graph. We substitute the expressions into the relation basis C − 3L, Q − 2L and omit the vertices which are not involved in cycles. The result which we have stated drops out.

In order to compute a discriminant, we have to choose a cutting set, e.g. the edges AC and CE. We replace A by ax, C by bx + cy, E by dy and obtain

(b − 6a)x + (c + 3d)y
−4ax + 3dy

and the resulting determinant is 3d(b − 2a) + 4ac. This is the discriminant of the corresponding tower. The multiplicities (a, b) = (c, d) = (1, 1) e.g. yield a homology plane.

(0.17) Remark. It is easy to verify, that cases G, K, M, and N do not give homology planes. Therefore we have not displayed the graphs K, M, and N. Also, in the other cases, not all cutting sets give homology planes. For instance, case A has (up to symmetry) three cutting sets, but for one of the cutting sets the discriminant has never absolute value one.

IV Chern numbers

In this section we compute the logarithmic Chern numbers of the homology planes which were constructed in section 2. Let X be a non-singular rational projective variety and D ⊂ X a normal crossing divisor with rational curves D_{1}, …, D_{u} as irreducible components. Let Γ(D) denote the dual graph of D. We use the following notation:

a_{0} number of path-components of Γ(D),

a_{1} number of cycles of Γ(D),

a_{2} number of vertices of Γ(D). Let K_{X} be a canonical divisor of X. The logarithmic Chern numbers of V = X\D are (defined as)

c̄_{2}(V) = e(X) − e(D) difference of Euler numbers,
c̄_{1}(V)^{2} = (K_{X} + D)^{2} self-intersection number.

Let b_{2} = b_{2}(X) denote the second Betti-number of X. From the homological and combinatorial definition of the Euler characteristic we obtain:

(0.18) {(1) e(X) = 2 + b_{2}(X). \atop (2) e(D) = a_{0} − a_{1} − a_{2},}

Next we compute (K_{X} + D)^{2} = K_{X}^{2} + (K_{X} + D) · D + K_{X} · D.

(0.19) Lemma.

(1) K_{X}^{2} = 10 − b_{2}(X).

(2) (K_{X} + D) · D = 2(a_{1} − a_{0}).

(3) K_{X} · D = −2a_{2} − ∑_{i} D_{i}^{2}.

Proof. (1) is true for X = IP^{2} and follows in general by using the change of K_{X}^{2} and b_{2}(X) under an expansion (Hartshorne [1977], V.3.3).

(2). For a smooth rational curve C the adjunction formula says (K_{X} +D)·C = −2 (Hartshorne [1977], V.1.5). Using this, (2) is proved by induction on the number of irreducible curves.

(3) is a rewriting of the adjunction formula. □

From (4.1) and (4.2) we obtain (compare Gurjar – Shastri [1989]):

(0.20)  (1) c̄_{2}(V) = 2 + b_{2} − a_{0} + a_{1} − a_{2}, ; (2) c̄_{1}(V)^{2} = 10 − b_{2} − 2a_{0} + 2a_{1} − 2a_{2} − ∑ D_{i}^{2}, ; (3) 3c̄_{2}(V) − c̄_{1}(V)^{2} = 4b_{2} − 4 − a_{0} + a_{1} − a_{2} + ∑ D_{i}^{2}.

We abbreviate the value (4.3.3) by p(V).

(0.21) Proposition. The varieties V which appear in towers of arrangements (A) – (N) have p(V) = 5.

Proof. One calculates p(V) by using (4.3.3) for the weighted graphs Figure (A)

– (N) and by noting that cutting a cycle increases the number p(V) by one. □

(0.22) Remark. For a homology plane V one always has c̄_{2}(V) = 3. Therefore (4.4) is equivalent to c̄_{1}(V)^{2} = −2 in the case of homology planes.

Note that (4.4) is a sharpening of the Miyaoka inequality (Miyaoka [1984], Theorem 1.1) for homology planes. This fact was also observed by Flenner and Zaidenberg. They relate it to the deformation theory of the surfaces. The result seems surprising. So one is tempted to state the following conjecture.

(0.23) Conjecture. For every homology plane V of general type the relation p(V) = 5 holds.

If a homology plane of general type is obtained from an arrangement of lines and quadrics one can translate p(V) = 5 into a combinatorial assertion about the arrangement. E.g., if V is obtained from an arrangement of n lines which has t_{r} points of multiplicity r, then the relation p(V) = 5 amounts to

(0.24)  ∑_{r≥2}(r − 2)t_{r} = 2n − 8.

V References

tom Dieck, T. [1990]: Symmetric homology planes. Math. Ann. 286, 143–152.

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

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

tom Dieck, T. and T. Petrie [1991]: Homology planes and algebraic curves. Preprint series Mathematica Gottingensis 15. To appear.

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

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

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

Matsuoka, T., and F. Sakai [1989]: The degree of rational cuspidal curves. Math. Ann. 285, 233–247.

Miyaoka, Y. [1984]: The maximal number of quotient singularities on surfaces with given numerical invariant. Math. Ann. 268, 159–171.

Namba, M. [1984]: Geometry of Projective Algebraic Curves. New York–Basel, Marcel Dekker.

Neusel, M. [1991]: Dissertation Göttingen.

Tammo tom Dieck Mathematisches Institut Bunsenstraße 3/5 3400 Göttingen Germany

Version November 18, 2000