Chern Classes of Fibered Products of Surfaces

Mina Teicher

Received: March 16, 1998 Revised: September 18, 1998

Communicated by Thomas Peternell

Abstract. In this paper we introduce a formula to compute Chern classes of fibered products of algebraic surfaces. For f : X → ℂℙ^{2} a generic projection of an algebraic surface, we define X_{k} for k ≤ n (n = deg f) to be the closure of k products of X over f minus the big diagonal. For k = n (or n − 1), X_{k} is called the full Galois cover of f w.r.t. full symmetric group. We give a formula for c_{1}^{2} and c_{2} of X_{k}. For k = n the formulas were already known. We apply the formula in two examples where we manage to obtain a surface with a high slope of c_{1}^{2}/c_{2}. We pose conjectures concerning the spin structure of fibered products of Veronese surfaces and their fundamental groups.

Keywords and Phrases: Surfaces, Chern classes, Galois covers, fibered product, generic projection, algebraic surface.

1991 Mathematics Subject Classification: 20F36, 14J10

0. Introduction. When regarding an algebraic surface X as a topological 4-manifold, it has the Chern classes c_{1}^{2}, c_{2} as topological invariants. These Chern classes satisfy:

c_{1}^{2}, c_{2} ≥ 0
5c_{1}^{2} ≥ c_{2} − 36
Signature = τ = \frac{1}{3}(c_{1}^{2} − 2c_{2})

The famous Bogomolov-Miyaoka-Yau inequality from 1978 (see [Re], [Mi], [Y]) states that the Chern classes of an algebraic surface also satisfy the inequality

c_{1}^{2} ≤ 3c_{2}.

It is known that this inequality is the best possible since Hirzebruch showed in 1958 that the equality is achieved by complex compact quotients of the unit ball (see [H]).

We want to understand the structure of the moduli space of all surfaces with given c_{1}^{2}, c_{2}; and, in particular, to populate it with interesting structures of surfaces. As a first step it is necessary to develop techniques to compute Chern classes of different surfaces.

In this paper we compute Chern classes of Galois covers of generic projections of surfaces. This was already computed in [MoTe2] for the case of the full Galois cover, where the product is taken n times (n is the degree of the projection). In this paper we deal with products taken k times, k < n, and we manage to give an example of a surface where the slope (c_{1}^{2}/c_{2}) is very high (up to 2.73). In subsequent research, using the results of this paper and of our ongoing research on this subject, we plan to further study these constructions, to compute these fundamental groups and to decide when the examples are spin, of positive index, etc. We conjecture that for X_{b} the Veronese surface of order b greater than 4, X_{k} is spin if k is even or b = 2, 3(4). We further conjecture that for the Hirzebruch surfaces in general the fundamental groups of X_{k} are finite.

In [RoTe], we used similar computations to produce a series of examples of surfaces with the same Chern classes and different fundamental groups which are spin manifolds where one fundamental group is trivial and the other one has a finite order which is increasing to infinity. The computations in this paper will lead to more examples of pairs in the τ > 0 area.

We consider in this paper fibered products and Galois covers of generic projections of algebraic surfaces. If f : X → ℂℙ^{2} is generic of deg n, we define the k-th Galois cover for k ≤ n to be \overline{X × ⋯ × X − Δ}_{f f} where Δ is the big diagonal and the fibered

product is taken k times. There exists a natural projection g_{k} : X_{k} → ℂℙ^{2}, deg g_{k} = n(n − 1) … (n − k + 1).

The surface X_{k} for k = n, is called the full Galois cover (i.e., the Galois cover w.r.t. full symmetric group), and is also denoted X_{Gal} or X̃. Clearly, deg(X_{Gal} → ℂℙ^{2}) = n!. It can be shown that X_{n} ≃ X_{n−1}. The full Galois covers were first treated by Miyaoka in [Mi], who noticed that their signature should be positive. In our papers [MoTe1], [MoTe2], [MoTe3], [MoRoTe], [RoTe], [Te], [FRoTe], we discussed the full Galois covers for X = f_{|aℓ_{1}+bℓ_{2}|}(ℂℙ^{1} × ℂℙ^{1}), Veronese embeddings and Hirzebruch surfaces. In the papers cited above we computed their fundamental groups (which are finite), the Chern numbers and the divisibility of the canonical divisor (to prove that when considered as 4-manifolds they are spin manifolds). X_{Gal} are minimal smooth surfaces of general type. Other examples of interest on surfaces in the τ > 0 area can be found in [Ch] and [PPX].

§1. The Main Theorem. We start with a precise definition.

Definition. A Galois cover of a generic projection w.r.t. the symmetric group S_{k} (for k < degree of the generic projection). Let X ↪ ℂℙ^{N} be an embedded algebraic surface. Let f : X → ℂℙ^{2} be a generic projection, n = deg f. For 1 ≤ k ≤ n, let

X × ⋯ ×_{f f} X = {(x_{1}, … x_{k}) | x_{i} ∈ X, f(x_{i}) = f(x_{j}) ∀i∀j},

Δ = {(x_{1}, …, x_{k}) ∈ X × ⋯ ×_{f f} X | x_{i} = x_{j} for some i ≠ j}

X_{k} = \overline{X × ⋯ ×} ; \underbrace{f f}_{k}\overline{X − Δ}.

X_{k} is the closure of X × ⋯ ×_{f f} X − Δ. X_{k} is the Galois cover w.r.t. the symmetric

group on k elements. We denote X_{0} = ℂℙ^{2}.

For every k ≥ 1 we have the canonical projections g_{k} : X_{k} → ℂℙ^{2} and a natural projection (on the first k factors) f_{k} : X_{k} → X_{k−1}, which satisfy

f_{1} = g_{1} = f
g_{k−1}f_{k} = g_{k}, (k ≥ 2).

Clearly,

deg g_{k} = n · (n − 1) … (n − k + 1)
deg f_{k} = n − k + 1,
X_{n−1} ≃ X_{n} (f_{n} is an isomorphism).

.

For k = n (or n − 1), we call X_{k} the Galois cover w.r.t. the full symmetric group or the full Galois cover and denote it also by X_{Gal}.

Remark. X_{k} is the interesting component in the fibered product X × ⋯ × X ; \underbrace{f f}_{k}

Notations.

For the rest of the paper we shall use the following notations:

n = deg f.

X_{k}, the Galois cover of f : X → ℂℙ^{2} as above, k ≤ n.

S = the branch curve of f in ℂℙ^{2} (S is a cuspidal curve)

m = deg S

μ = deg S^{∗} (S^{∗} the dual to S)

= number of branch points in S w.r.t. a generic projection of ℂ^{2} to ℂ^{1}. d = number of nodes in S ρ = number of cusps in S Theorem 1. The Chern classes of X_{k} are as follows:

(a) c_{1}^{2}(X_{1}) = 9n + (\frac{m}{2} − 6) m − ρ − d.

For 2 ≤ k ≤ n − 1

c_{1}^{2}(X_{k}) = 9(n − k + 1) … n

+ \frac{1}{2}[(n − k + 1) … (n − 2)](2n − k − 1)k (\frac{m}{2} − 6) m

− [(n − k − 1) … (n − 3)]kρ

− \frac{1}{2}[(n − k − 1) … (n − 4)](2n − k − 5)kd

(b)

c_{2}(X_{1}) = 3n − 2m + μ
c_{2}(X_{2}) = 3n(n − 1) − 2(2n − 3)m + (2n − 3)μ + ρ + 2d
c_{2}(X_{3}) = 3n(n − 1)(n − 2) − 3(2n − 4)(n − 2)m + \frac{3}{2}(2n − 4)(n − 2)μ
+ 2(3n − 9)d + (3n − 8)ρ

For 4 ≤ k ≤ n − 1

c_{2}(X_{k}) = 3(n − k + 1) … n

− (n − k + 1) … (n − 2)(2n − k − 1)km

+ \frac{1}{2}(n − k + 1) … (n − 2)(2n − k − 1)kμ

+ (n − k + 1) … (n − 3)(k − 1)k (\frac{n}{2} − \frac{k + 1}{3}) ρ

+ [(n − k + 1) … (n − 4)]\frac{k(k + 1)}{4}\{(k + 6)(k − 1) + 4n(n − k − 1)\}d

+ [(n − k + 1) … (n − 4)]{4nk − 2n^{2}k}d

Remarks.

(a) We consider an empty multiplication as 1.

(b) The case k = n − 1(X_{k} = X_{Gal}), of this Theorem was treated in [MoTe2], Proposition 0.2 (proof there is given by F. Catanese). (See also [MoRoTe]). One can easily see that for k = n the formulas here coincide with the formulas from [MoTe2]. For c_{1}^{2} it is enough to use remark (a) about empty multiplication. We get:

c_{1}^{2}(X_{Gal}) = c_{1}^{2}(X_{n−1}) = \frac{n!}{4}(m − 6)^{2}.

Note that d and ρ do not appear in this formula. For c_{2} we get here (using (a_{1}))

c_{2}(X_{Gal}) = c_{2}(X_{n−1}) = n! (3 − m + \frac{1}{4}d + \frac{μ}{2} + \frac{ρ}{6})

which coincide with [MoTe2], using the formula for the degree of the dual curve:

μ = m^{2} − m − 2d − 3ρ.

Proof of the Theorem.

Let g_{k} : X_{k} → ℂℙ^{2}, f_{k} : X_{k} → X_{k−1} be the natural projections. Clearly, g_{1} =

f_{1} = f, g_{k} = g_{k−1}f_{k}, for k ≥ 2 deg f_{k} = n − k + 1, deg g_{k} = \frac{n!}{(n − k)!}. Let E_{k} and

K_{X_{k}} be the hyperplane and canonical divisors of X_{k}, respectively (E_{k} = g^{∗}(ℓ) for a line ℓ in ℂℙ^{2}).

Let S_{k} be the branch curve of f_{k} (in X_{k−1}), m_{k} its degree and μ′_{k} the number of branch points that do not come from S_{k−1} (S_{1} = S). Let S′_{k} be the ramification locus of f_{k} (in X_{k}). Let T′_{k} be the ramification locus of g_{k} (in X_{k}).

We recall that the branch points in S (or S_{k}) come from two points coming together in the fibre, the cusps from (isolated) occurrences of three points coming together and nodes from 4 points coming together into 2 distinct points. Generically, cusps and nodes are unbranched. We use this observation in the sequel.

To compute c_{1}^{2}(X_{k}) we shall use:

c_{1}^{2}(X_{k}) = K_{X_{k}}^{2}
K_{X_{k}} = −3E_{k} + T′_{k}

and the following identities.

T′_{k} = \{ \begin{matrix}S′_{k} + f_{k}^{∗}(T′_{k−1})  k ≥ 2 \\ S′_{1}  k = 1 \end{matrix}

T′_{k} = −\frac{1}{2}S_{k+1} + \frac{1}{2}g_{k}^{∗}(S)

To compute c_{2}(X_{k}) we shall assume that all cusps and nodes of S are vertices of a triangulation. Using the standard stratification computations, this implies the following recursive formula:

c_{2}(X_{k}) = deg f_{k} · c_{2}(X_{k−1}) − 2m_{k} + μ′_{k}.

Thus we need to get a formula for E_{k} · T′_{k}, S_{k+1} · T′_{k}, m_{k} and μ′_{k}. We shall use the following 3 claims:

Claim 1.

(i) Let m_{k} = deg S_{k}. For k ≥ 2, m_{k} = (n − k) … (n − 2)m, m_{1} = m.

(ii) Let d_{k} = # nodes in S_{k}. For k ≥ 2, d_{k} = (n − k − 2) … (n − 4)d, d_{1} = d. (iii) Let ρ_{k} = # cusps in S_{k}. For k ≥ 2, ρ_{k} = (n − k − 1) … (n − 3)ρ, ρ_{1} = ρ. (iv) Let μ′_{k} be the number of branch points of S_{k} that do not come from S_{k−1},

μ′_{k} = μ_{k} −(n−k +1)μ_{k−1} (k ≥ 2) and μ′_{1} = μ. Then for k = 2, μ′_{2} = (n−2)μ+ρ+2d and for k ≥ 3 μ′_{k} = (n − k) … (n − 2)μ+ (n − k) … (n − 3)(k − 1)ρ + [(n − k) … (n − 4)](k − 1)(2n − k − 4)d. (For k = 3 the coefficient of d is 2(2n − 7).)

Claim 2.

E_{k.}T′_{k} = \{ \begin{matrix}m & k = 1 \\ \frac{1}{2}m[(n − k + 1) … (n − 2)]\{(2n − 1)k − k^{2}\} & k ≥ 2 \end{matrix}

Claim 3.

S_{k+1.}T′_{k} = \{ \begin{matrix}2ρ + 2d & k = 1 \\ 2(n−k−1) … (n−3)kρ + (n−k−1) … (n−4)(2n−k−5)kd & k ≥ 2 \end{matrix}

Proof of Claim 1.

Items (i), (ii) and (iii) are easy to verify from the definition of fibered product. For (iv) we notice that {μ′_{k}} satisfy the following recursive equations:

μ′_{k} = (n − k)μ′_{k−1} + ρ_{k−1} + 2d_{k−1} k ≥ 2
μ′_{1} = μ.

The formula for μ′_{2}, μ′_{3} follows immediately from the recursive formula. For k ≥ 4 we substitute the formulas for ρ_{k−1} and d_{k−1} from (ii) and (iii) to get μ′_{k} = (n − k)μ′_{k−1} + (n − k) … (n − 3)ρ + 2(n − k − 1) … (n − 4)d and we shall proceed by induction. By the induction hypothesis μ′_{k−1} = (n − k + 1) … (n − 2)μ + (n − k + 1) … (n − 3)(k − 2)ρ + [(n − k + 1) … (n − 4)](k − 2)(2n − k − 3)d. We substituted the last expression in the previous one to get

μ′_{k} = (n − k)(n − k + 1) … (n − 2)μ + (n − k)(n − k + 1) … (n − 3)(k − 2)ρ
+ (n − k)(n − k + 1) … (n − 4)(k − 2)(2n − k − 3)d
+ (n − k) … (n − 3)ρ + 2(n − k − 1) … (n − 4)d
= (n − k) … (n − 2)μ + (n − k) … (n − 3)(k − 1)ρ
+ (n − k) … (n − 4){(k − 2)(2n − k − 3) + 2(n − k − 1)}d

which coincide with the claim since (k − 2)(2n − k − 3) + 2(n − k − 1) = (k − 1)(2n − k − 4). □ for Claim 1

Proof of Claim 2.

For k ≥ 2

E_{k.}T′_{k} = \frac{1}{2}E_{k.}(g_{k}^{∗}(S) − S_{k+1})

= \frac{1}{2}E_{k}g_{k}^{∗}(S) − \frac{1}{2}E_{k.}S_{k+1}

= \frac{1}{2}g_{k}^{∗}(ℓ)g_{k}^{∗}(S) − \frac{1}{2}E_{k.}S_{k+1}

= \frac{1}{2}g_{k}^{∗}(ℓ.S) − \frac{1}{2}E_{k.}S_{k+1}

= \frac{1}{2}(deg g_{k.})m − \frac{1}{2}m_{k+1}

= \frac{1}{2}m(n − k + 1) … n − \frac{1}{2}(n − k − 1)(n − k) … (n − 2)m

= \frac{1}{2}m[(n − k + 1) … (n − 2)]\{(n − 1)n − (n − k − 1)(n − k)\}

= \frac{1}{2}m[(n − k + 1) … (n − 2)]\{2nk − k − k^{2}\}. □ for Claim 2

Proof of Claim 3.

Since T′_{1} = S′_{1}, the formula trivializes for k = 1. S_{2} · T′_{1} = S_{2} · S′_{1} = 2ρ + 2d = 2ρ_{1} + 2d_{1}. For k ≥ 2

S_{k+1} · T′_{k} = S_{k+1}(f_{k}(T′_{k−1})′ + S′_{k})
= S_{k+1} · f_{k}^{∗}(T′_{k−1}) + (S_{k+1} · S′_{k})
= (deg f_{k} |_{S_{k+1}}) · (S_{k} · T′_{k−1}) + 2ρ_{k} + 2d_{k}
= (deg f_{k} − 2)(S_{k} · T′_{k−1}) + 2ρ_{k} + 2d_{k}
= (n − k − 1)(S_{k} · T′_{k−1}) + 2ρ_{k} + 2d_{k}.

Denote a_{k} = S_{k+1} · T′_{k}.

We shall prove the claim by induction using the recursive formula a_{k} = (n − k − 1)a_{k−1} + 2ρ_{k} + 2d_{k}. For k = 2 :

a_{2} = (n − 3)a_{1} + 2ρ_{2} + 2d_{2}
= (n − 3)(2ρ + 2d) + 2(n − 3)ρ + 2(n − 4)d
= 4(n − 3)ρ + 2d(n − 3 + n − 4)
= 4(n − 3)ρ + 2d(2n − 7).

Thus the statement is true for k = 2.

Let k ≥ 3. Assume the formula is true for k − 1. We shall prove it for k.

a_{k} = (n−k−1)a_{k−1} + 2ρ_{k} + 2d_{k}

= (n−k−1){2(n−k) … (n−3)(k−1)ρ + (k−1)(n−k) … (n−4)(2n−k−4)d} + 2(n−k−1) … (n−3)ρ + 2(n−k−2) … (n−4)d = 2(n−k−1) … (n−3)kρ + (n−k−1) … (n−4){(2n−k−4)(k−1) + 2(n−k−2)}

= 2(n−k−1) … (n−3)kρ + (n−k−1) … (n−4){2nk−k^{2}−5k}d.

From the two formulae, we can see that the product (n − k − 1)...(n − 4) should be 1 for k ≤ 2. □ for Claim 3

We go back to the proof of the theorem. To prove (a) we write

c_{1}^{2}(X_{k}) = K_{X_{k}}^{2} = (−3E_{k} + T′_{k})^{2}

= 9E_{k}^{2} − 6E_{k}.T′_{k} + (T′_{k})^{2}

= 9E_{k}^{2} − 6E_{k.}T′_{k} + T′_{k} [−\frac{1}{2}S_{k+1} + \frac{1}{2}g_{k}^{∗}(S)]

= 9E_{k}^{2} − 6E_{k} · T′_{k} − \frac{1}{2}T′_{k} · S_{k+1} + \frac{1}{2}T′_{k} · g_{k}^{∗}(S).

Now: E_{k}^{2} = deg g_{k} = (n−k+1) … n (= \frac{n!}{(n − k)!}) . Since S is of deg m T′_{k}.g_{k}^{∗}(S) =

mE_{k}.T′_{k}. We substitute the results from Claim 2 and Claim 3 to get (a).

We prove (b) by induction on k. For k = 1 we take the recursive formula c_{2}(X_{k}) = (n − k + 1)c_{2}(X_{k−1}) − 2m_{k} + μ′_{k} and substitute k = 1 to get c_{2}(X_{1}) = 3n − 2m + μ which coincides with formula (b) for k = 1. We do the same for k = 2, 3. To prove k − 1 implies k we use Claim 1(iv) and (ii) to write

c_{2}(X_{k}) = (n − k + 1)c_{2}(X_{k−1}) − 2(n − k) … (n − 2)m + (n − k) … (n − 2)μ
+ (k − 1)(n − k) … (n − 3)ρ
+ (n − k) … (n − 4)(k − 1)(2n − k − 4)d

When substituting the inductive statement for c_{2}(X_{k−1}) and shifting around terms, we get (b).

□ for the Theorem

§2. A Different Presentation of the Chern Classes.

Proposition 2. Let E and K denote the hyperplane and canonical divisors of X, respectively. Then the Chern classes of X_{k} are functions of c_{1}^{2}(X), c_{2}(X), deg(X), E, K, and k.

Proof. (Proof for X_{n} appeared in [RoTe]) Let S be the branch curve of the generic projection f : X → ℂℙ^{2} (S ⊆ ℂℙ^{2}). By Theorem 1, the Chern classes of X_{k} depend on k, deg(S), deg(X) and μ, d, ρ, the number of branch points, nodes and cusps of S, respectively.

We shall first show that μ, d, ρ depends on c_{2}(X), deg X, deg(S), e(E) and g(R) where g denotes the genus of an algebraic curve, e denotes the topological Euler characteristic of a space, and R (⊂ X) is the ramification locus of f which is, in fact, the non-singular model of S.

Recall that μ also is equal to deg(S^{∗}), where S^{∗} is the dual curve to S. For short we write n = deg(X), m = deg(S).

We show this by presenting three linearly independent formulae:

μ = m(m − 1) − 2d − 3ρ

g(R) = \frac{(m − 1)(m − 2)}{2} − d − ρ

c_{2}(X) + n = 2e(E) + μ

The first two are well-known formulae for the degree of the dual curve and the genus of a non singular model of a curve. For the third, we may find a Lefschetz pencil of hyperplane sections of X whose union is X. The number of singular curves in the pencil is equal to μ. The topological Euler characteristic of the fibration equals e(X) = e(ℂℙ^{1}) · e(E) + μ − n (n appears from blowing up n points in the hyperplane sections). The formula follows from e(ℂℙ^{1}) = 2 and e(X) = c_{2}(X).

We shall conclude by showing that deg(S), e(E) and g(R) depend on c_{1}^{2}(X), deg X and E.K.

This follows from the Riemann-Hurwitz formula, R = K + 3E, the adjunction formula 2 − 2g(C) = −C.(C + K), and the fact that E^{2} = deg X and K^{2} = c_{1}^{2}(X). In fact, we have:

g(R) = 1 + \frac{1}{2}R(R + K) = 1 + \frac{1}{2}(K + 3E)(2K + 3E)

e(E) = 2 − 2g(E) = −E(E + K)

deg(S) = deg(R) = E.R = E(K + 3E). □

From the above proof we can, in fact, get the precise formulae of c_{1}^{2}(X_{k}) and c_{2}(X_{k}) in terms of c_{1}^{2}(X), c_{2}(X), deg(X), E.K, and k. For certain (computerized) computations, it is easier to work with these formulae rather than those of Theorem 1.

Corollary 2.1. In the notations of the above proposition:

c_{1}^{2}(X_{n}) = \frac{n!}{4}[(E.K)^{2} + 6n(E.K) + 9n^{2} − 12(E.K) − 36n + 36]

c_{2}(X_{n}) =

\frac{n!}{24}[72 − 10c_{1}^{2}(X) − 54(E.K) − 114n + 27n^{2} + 14c_{2}(X) + 3(E.K)^{2} + 18n(E.K)]

Similar formulas can be obtained for X_{k} for k < n.

§3. Examples. To use Theorem 1, we need computations of n, m, μ, ρ and d. We compute them for two examples.

Examples 3.1. For X = V_{b}, a Veronese embedding of order b, we have

n = b^{2}

m = 3b(b − 1)

μ = 3(b − 1)^{2}

φ = 3(b − 1)(4b − 5)

d = \frac{3}{2} (b − 1)(3b^{3} − 3b^{2} − 14b + 16)

(see [MoTe3]).

Proof. For n, m, μ and ρ, see [MoTe3] and [MoTe4]. Since μ = m(m − 1) − 2d − 3ρ, we get the following formula for d: 2d = m^{2} − m − μ − 3φ and thus

2d = 3b(b − 1)(3b(b − 1) − 1) − 3(b − 1)^{2} − 9(b − 1)(4b − 5)
= 3(b − 1){(3b^{2} − 3b − 1)b − (b − 1) − 3(4b − 5)}
= 3(b − 1){3b^{3} − 3b^{2} − 14b + 16}.

When one substitutes b = 3 and k = 4, one gets \frac{c_{1}^{2}}{c_{2}} = 2.73. By experimental substitutions it seems that for large b, the signature τ(X_{k}) (= c_{1}^{2} − 2c_{2}), changes from negative to positive at about \frac{3}{4}n.

Example 3.2. For X = X_{t(a,b)} = f_{|aℓ+bC_{+}|} (Hirzebruch surface of order t), where ℓ is a fiber, (C_{+})^{2} = t, and a ≥ 1, we have

n = 2ab + tb^{2}
m = 6ab − 2a − 2b + t(3b^{2} − b)
μ = 6ab − 4a − 4b + 4 + t(3b^{2} − 2b)
φ = 24ab − 18a − 18b + 12 + t(12b^{2} − 9b)

Proof. [MoRoTe], Lemma 7.1.3.

Example 3.3. (in the τ < 0 area) For X a K3 surface:

K = 0

c_{1}^{2}(X) = K^{2} = 0

c_{2}(X) = 24

n = 4

m = 12

μ = 36

ρ = 24

d = 12

c_{1}^{2}(X_{2}) = 48

c_{2}(X_{2}) = 144

c_{1}^{2}(X_{3}) = c_{1}^{2}(X_{Gal}) = 216

c_{2}(X_{3}) = c_{2}(X_{Gal}) = 240.

Proof. It is well known that for a K3 surfaces K = 0, c_{1}^{2} = 0, c_{2} = 24, S′ = 3E, n = E^{2} = 4. Using this we can get m and μ:

m = S′.E = 3E.E = 3E^{2} = 12
μ = c_{2}(X) − 2e(E) + n (see proof of Proposition 2)
= c_{2}(X) − 2(2 − 2g(E)) + n
= c_{2}(X) + 2E.(E + K) + n = 36

Now from

m(m − 1) = μ + 3ρ + 2d

and

m(m − 3) = 2g(S′) − 2 + 2ρ + 2d = (K + S′), S′ + 2ρ + 2d
= 3E.3E + 2ρ + 2d = 9E^{2} + 2ρ + 2d,

we get 2m = μ − 9E^{2} + ρ = 24 and ρ = 2m + 9E^{2} − μ = 24.

Moreover, we get d = \frac{1}{2}(m(m − 1) − μ − 3ρ) = 12. We substitute these quantities in the formula from Theorem 1 to get the values of the Chern classes.

Remark. For t = 0, X_{t(a,b)} are actually embeddings of ℂℙ^{1} × ℂℙ^{1}. In [FRoTe], we computed the fundamental group of X_{n} = X_{Gal} for X = X_{t(a,b)} which is ℤ_{c}^{n−2} for c = g.c.d.(a, b). Thus for (a, b) = 1 these surfaces are simply connected. All these surfaces are smooth minimal surfaces of general type. For a ≥ 6, b ≥ 5, the signature of these surfaces is positive. For five pairs of (a, b), these surfaces have signature 0 (see [MoRoTe]). Four of these surfaces are simply connected and the fifth one for which a = b = 5, π_{1}(X_{Gal}) = ℤ_{5}^{48}.

In our ongoing research, we shall apply Theorem 1 and Proposition 2 in order to obtain more examples of non diffeomorphic surfaces or surfaces in different deformation families with the same c_{1}^{2} and c_{2}, as well as to compute the slope \frac{c_{1}^{2}}{c_{2}} and to search for higher slopes.

We are also interested in the fundamental groups (in particular, in the finite ones) and the divisibility of the canonical class (in particular, the case where the canonical class is divided by 2, i.e., the spin case), which we will investigate in a subsequent paper. The results in this paper are a basis for producing interesting examples of surfaces with positive index, (c_{1}^{2} − c_{2}), finite fundamental groups and spin (K even) structure. In particular, we plan to prove the following two conjectures.

Conjecture. For X = V_{b}, Veronese of order b, b > 4, we have X_{k} is a spin manifold ⇔ k even or b = 2, 3(4).

Conjecture. For X = F_{t(a,b)} (the Hirzebruch surface), π_{1}(X_{k}) is finite.

References

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[FRoTe] Freitag P., Robb A., Teicher M., The fundamental group of Galois covers of Hirzebruch surfaces, in preparation.

[H] Hirzebruch F., Automorphe Formen und der Satz von Riemann-Roch, Internat. Sympos. on Algebraic Topology, Univ. Nacional Autonoma de Mexico and UNESCO, Mexico City (1958), 129-144.

[Mi] Miyaoka M., On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 239-272.

[MoRoTe] Moishezon B., Robb A., Teicher M., On Galois covers of Hirzebruch surfaces, Math. Ann. 305 (1996), 493-539.

[MoTe1] Moishezon B., Teicher M., Existence of simply connected algebraic surfaces of positive and zero indices, Proceedings of the National Academy of Sciences, United States of America 83 (1986), 6665-6666.

[MoTe2] Moishezon B., Teicher M., Simply connected algebraic surfaces of positive index, Invent. Math. 89 (1987), 601-643.

[MoTe3] Moishezon B., Teicher M., Galois coverings in the theory of algebraic surfaces, Proc. of Symp. in Pure Math. 46 (1987), 47-65.

[MoTe4] Moishezon B., Teicher M., Braid group techniques in complex geometry III, Projective degeneration of V_{3}, Contemporary Math. 162 (1994), 313-332.

[PPX] Persson U., Peters C., Xiao G., Geography of spin manifolds, Math. Ann. (to appear).

[Re] Reid M., On the Bogomolov’ Theorem, Preprint, 1976.

[RoTe] Robb A., Teicher M., Applications of braid group techniques to the decomposition of moduli spaces, new examples, Topology and its Applications 78 (1997), 143-151.

[Te] Teicher M., Braid groups, algebraic surfaces and fundamental groups of complements of branch curves, Amer. Math. Soc. Publications PSPM 62 (1997), no. 1, 127-150.

[Y] Yau S.-T., Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. USA 74 (1977), 1798-1799.

Mina Teicher Emmy Noether Research Institute for Mathematics. Bar-Ilan University Ramat-Gan, 52900 Israel teicher@macs.biu.ac.il