Diagonal Quotient Surfaces

E. Kani* and W. Schanz

Introduction

Let X be a smooth projective curve over ℂ which admits a finite group G of automorphisms. Then G × G acts on the product surface Y = X × X, and so for each subgroup H ≤ G × G, the quotient Z = H\Y is a normal algebraic surface. Here we shall restrict attention to the case that the subgroup H is the graph of a group automorphism α ∈ Aut(G) of G, i.e. H = Δ_{α} = {(g, α(g)) : g ∈ G}; we propose to call the resulting quotient surface a (twisted) diagonal quotient surface, and denote it by

Z_{X,G,α} = Δ_{α}\Y.

Such surfaces occur naturally as the (coarse) moduli spaces of certain moduli problems; cf. [17]. In this case X = X(N), the modular curve of level N, and there is a close analogy between these surfaces and the Hilbert modular surfaces studied by Hirzebruch and others (cf. van der Geer[6]).

The main objective of this paper is to analyze the geometry of the diagonal quotient surfaces Z_{X,G,α} and of their desingularizations Z̃_{α} by calculating their numerical invariants such as their Betti and Chern numbers, and to determine their place in the Enriques-Kodaira classification table.

One might expect that all these invariants can be expressed in terms of data involving the curve X and the group G, and this indeed turns out to be the case. In fact, they can all be expressed in terms of the genus g = g(X) of X, the genus ḡ = g(X̄) of the quotient curve X̄ = G\X, and three fundamental invariants 𝔾_{α}, 𝕊_{α} and 𝕃_{α} which are essentially of a local nature in that they depend on the local action of G at the fixed points. For example, we have the following result which is a special case of Theorem 3.7 below:

Theorem 1 The irregularity, the geometric genus and the square of the first Chern class of the desingularization Z̃_{α} of the diagonal quotient surface Z_{α} are given by:

q = 2ḡ, p_{g} = 𝔾_{α} − 𝕊_{α}, and c_{1}^{2} = 8(1 − 2ḡ + 𝔾_{α}) − 𝕃_{α} − 12𝕊_{α}.

*NSERC University Research Fellow

Of course, the above result is only useful if we succeed in calculating the invariants 𝔾_{α}, 𝕃_{α} and 𝕊_{α}. Of these, 𝔾_{α} is perhaps the simplest since it can be expressed in terms of the genera g_{x̄,α} of certain subquotients X_{x̄,α} of X or, alternatively, as the inner product (h_{α}^{1}, h^{1}) of the character h^{1} afforded by the G-module H^{1}(X, ℂ) (cf. Proposition 1.8). In a similar way, 𝕊_{α} can be expressed either as a weighted sum of Dedekind sums 𝕊(q, n) or in terms of certain inner products involving the character ω of the G-module H^{0}(X, Ω_{X}^{1}) of holomorphic differentials (cf. Theorem 1.13). Finally, the invariant 𝕃_{α}, which is closely related to the nature of the singularities of Z_{α}, is defined in terms of a weighted sum of lengths of certain continued fractions; cf. Proposition 3.5. In both cases, these weights are given by the “local invariants” s_{ν}(x̄, ȳ) which are introduced in section 1.1.

From the knowledge of the Chern numbers it is often possible to determine the Kodaira dimension κ and hence the type of the surface according to the Enriques– Kodaira classification scheme. In particular, we have a complete classification in the case that either g ≤ 1 or that ḡ ≥ 1, as the following two theorems show (cf. Theorems 4.1 and 4.2):

Theorem 2 a) If g = 0 then Z̃_{α} is rational; i.e. κ = −∞.

b) If g = ḡ = 1, then Z̃_{α} ≃ Z_{α} is an abelian variety, so κ = 0.

c) If g = 1 and ḡ = 0, then Z̃_{α} is an elliptic surface with κ ≤ 0.

d) If g ≥ 2 and ḡ ≥ 1, then Z̃_{α} is a minimal surface of general type, i.e. κ = 2.

Theorem 3 If g = 1 and ḡ = 0, then p_{g} ≤ 1 and c_{1}^{2} ≤ 0, and we have the following three possibilities:

a) If p_{g} = 0 and c_{1}^{2} < 0, then Z̃_{α} is rational (κ = −∞).

b) If p_{g} = 0 and c_{1}^{2} = 0, then Z̃_{α} is a minimal Enriques surface (κ = 0).

c) If p_{g} = 1, then Z̃_{α} is a minimal K3-surface (κ = 0).

It should be remarked that all these possibilities actually do occur (cf. section 4.1).

The remaining cases (i.e. g ≥ 2 and ḡ = 0) are more subtle in that the Chern numbers themselves are often insufficient in determining the type of the surface, as examples with modular curves show. Nevertheless, one might expect that if m = |G| is small with respect to g, then Z̃_{α} is of general type. Theorem 4.8 gives some evidence for this expectation; for example we have:

Theorem 4 If there are at least r ≥ 8 points of X̄ = G\X which are ramified in X, then Z̃_{α} is of general type.

In order to be able to classify the surfaces Z̃_{α} in other cases, additional information seems to be necessary. Since it is difficult to calculate the plurigenera P_{n} of Z̃_{α}, we rely instead on the existence of certain curves and/or curve configurations on Z̃_{α}. To this end we introduce two criteria which are useful for distinguishing between elliptic surfaces and surfaces of general type. The first is based on the invariant d(S, C) = 2p_{g}(S) − 2 − (K_{S}.C) of a curve C lying on a regular surface S, and the second studies what we call (−2)-joins of two curves C_{1} and C_{2} on S; the latter are divisors D = C_{1} + C_{2} + E_{1} + … + E_{t} formed by the (−2)-chains E_{i} which join C_{1} to C_{2} (cf. Definition 4.16). More precisely, we have results of the following type (cf. Theorem 4.12 and Proposition 4.17):

Theorem 5 Let S be a smooth regular surface with κ(S) ≥ 1.

a) If S has a curve C with p_{a}(C) ≥ 2 and d(S, C) ≥ 1, then S is of general type.

b) If S has a (−2)-join D with D^{2} > (K_{S}.D)^{2}, then S is an elliptic surface.

Applying Theorem 5 to the curve configurations on Z̃_{α} arsing from the resolution of singularities leads to quite explicit criteria. To state these, let x ∈ X be point with non-trivial stabilzer group G_{x} ≤ G (i.e. x is a ramification point of π : X → X̄ = G\X), and let g_{x,α} = g(X_{x,α}) denote the genus of the quotient curve X_{x,α} = α^{−1}(G_{x})\X, r_{1,x,α} the number of unramified points above x̄ = π(x) in the covering π_{x,α} : X_{x,α} → X̄, and let

c_{x,α} = ∑_{ȳ∈X̄} ∑_{ν=1}^{e_{y}−1} s_{ν,α}(x̄, ȳ)\frac{ν}{eȳ} and s_{x,α} = ∑_{d|e_{x},d≠e_{x}} s_{e_{x}−d,α}(x̄, x̄),

where, as before, s_{ν,α}(x̄, ȳ) denotes the local invariant defined in section 1. (Note that g_{x̄,α} and r_{1,x,α} may also be expressed in terms of these local invariants; cf. Remark 1.5.) Then the aforementioned criteria may be formulated as follows (cf. Proposition 4.19):

Theorem 6 Suppose that ḡ = 0.

a) If g_{x,α} = g_{x,α^{−1}} = 0, c_{x,α} = c_{x,α^{−1}} = 1 and r_{1,x,α} > 0 for some (ramified) point x ∈ X, then Z̃_{α} is rational.

b) If we have

s_{x,α} > \frac{1}{2}(k_{x,α}^{2} − i_{x,α})

for some x ∈ X, where

k_{x,α} = 2g_{x,α} + 2g_{x,α^{−1}} + c_{x,α} + c_{x,α^{−1}} − 4 and i_{x,α} = 2r_{1,x,α} − c_{x,α} − c_{x,α^{−1}},

then Z̃_{α} is not of general type, i.e. κ(Z̃_{α}) ≤ 1. If, in addition, p_{g}(Z̃_{ε}) = 1 and g_{x,α} = g_{x,α^{−1}} = 0 and c_{x,α} = c_{x,α^{−1}} = 2, then Z̃_{α} is a (blown-up) K3-surface.

c) Suppose that κ(Z̃_{α}) ≥ 1 and that for some x ∈ X we have

2p_{g}(Z̃_{α}) > 2g_{x,α} + c_{x,α}.

If either g_{x,α} ≥ 2 or g_{x,α} = g_{x,α} = 1 and r_{1,x,α} > 0, then Z̃_{α} is of general type.

In [17], the above theorem will be used to determine the Enriques-Kodaira classification of the modular diagonal quotient surfaces Z̃_{α} = Z̃_{X,G,α} with X = X(N), the modular curve. It is interesting to note that all except two of the cases considered there are actually covered by the above theorem.

Acknowledgements: The authors are very thankful for helpful and stimulating discussions with G. Frey, A. Geramita, E. Viehweg, R. Weissauer, and J. Zarhin. The second author wants to thank Queen’s University for its hospitality during a visit supported by NSERC and ARC research grants held by the first author.

This research was supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) and also by the Advisory Research Council (ARC) of Queen’s University.

1 Curves with a group action

1.1 The local invariants

Let X be a smooth compact complex curve of genus g = g(X) and let G be a finite group of order m which acts faithfully on X by holomorphic automorphisms. Then the quotient X̄ = G\X is again a smooth compact curve, and the quotient map π : X → X̄ has degree deg(π) = m.

More generally, let H ≤ G be any subgroup of G, and let X_{H} denote the quotient of X with respect to H. We then have two associated coverings,

π_{H} : X → X_{H} and π̄_{H} : X_{H} → X̄,

of degree deg(π_{H}) = n_{H} := |H| and deg(π̄_{H}) = m/n_{H}, respectively, whose composition is π = π̄_{H} ∘ π_{H}. For future reference, we record the following elementary facts.

Remark 1.1 a) If x ∈ X is any point, then its stabilizer H_{x} = {τ ∈ H : τx = x} is always a cyclic subgroup of H, and we have H_{τx} = ^{τ}H_{x} = H^{τ^{−1}} = τH_{x}τ^{−1}, for any τ ∈ H. Its order is the ramification index of π_{H} at x: e_{x}(H) = e_{x}(π_{H}) = |H_{x}|. Since e_{x}(H) depends only on x̃ = π_{H}(x) ∈ X_{H} (see b)), we shall often write ex̃(H) = e_{x}(H).

b) Since the group H acts transitively on the points of the fibre π_{H}^{−1}(π_{H}(x)), the map τ ↦ τx induces a bijection

f_{x} = f_{H,x} : H/H_{x} →^{∼} π_{H}^{−1}(π_{H}(x))

between the right coset space H/H_{x} and the fibre at π_{H}(x).

c) Similarly, the map τ ↦ π_{H}(τx) induces a bijection

f̄_{x} = f̄_{H,x} : H\G/G_{x} →^{∼} X_{H,x̄} := π̄_{H}^{−1}(x̄)

between the double coset space H\G/G_{x} and the fibre X_{H,x̄} over x̄ = π(x). In this case the ramification index of y = f̄_{x}(HgG_{x}) ∈ π̄_{H}^{−1}(x̄) is given by

ē_{y}(H) := e_{y}(π̄_{H}) = \frac{|G_{x}|}{|H ∩ gG_{x}g^{−1}|} = \frac{|HgG_{x}|}{n_{H}}.

d) For any σ ∈ G there is a unique isomorphism σ_{H} : X_{H} →^{∼} X_{^{σ}H} such that π_{^{σ}H} ∘ σ = σ_{H} ∘ π_{H}. Since (^{σ}H)_{σx} = H_{x} for any x ∈ G, it follows that e_{σx}(^{σ}H) = e_{x}(H) and that e_{σ_{H} x̄}(^{σ}H) = ex̄(H), for x̄ ∈ X_{H}.

e) The map π_{H} is ramified at the set T_{H} = {x ∈ X : e_{x}(H) ≠ 1} of fixed points of H. Since e_{x}(H) = ex̄(H) depends only on x̄ = π_{H}(x) ∈ X_{H}, it follows that T_{H} = π_{H}^{−1}(R_{H}), where R_{H} = {P̄ ∈ X_{H} : eP̄(H) ≠ 1} denotes the ramification locus of π̄_{H}. Let us denote the cardinalities of these sets by r_{H} = |R_{H}| and t_{H} = |T_{H}|. Then by the Riemann–Hurwitz relation, the genus g_{H} = g(X_{H}) of X_{H} can expressed in terms of r_{H} and t_{H} as follows:

g_{H} = 1 + \frac{g − 1}{n_{H}} − \frac{1}{2} ∑_{x̃∈X_{H}} (1 − \frac{1}{ex̃} ) = 1 + \frac{g − 1}{n_{H}} − \frac{r_{H}}{2} + \frac{t_{H}}{2n_{H}} . (1)

In addition to the ramification indices e_{x}, there are further invariants attached to the covering π which will be of paramount importance in the study of diagonal quotient surfaces. These arise from the so-called “local action” of the stabilizer G_{x} at a fixed point x ∈ T = T_{G}. We briefly recall this notion and thereby also fix some notation.

Notation 1.2 For each (fixed) point x ∈ X, we have a canonical linear action of the stabilizer G_{x} on the cotangent space 𝔪_{x}/𝔪_{x}^{2} ≃ ℂ at x, which gives rise to a linear character λ_{x} = λ_{G,x} : G_{x} → ℂ. In terms of a local parameter (or coordinate) t = t_{x}, this character is given by:

τ^{∗}t ≡ λ_{x}(τ)t (mod 𝔪_{x}^{2}), for τ ∈ G_{x}. (2)

Alternatively, λ_{x} could also be described as follows. By a more judicious choice of the local coordinate t = t_{x} , we can arrange that the action is locally linear on X, so that we have τ^{∗}t = λ_{x}(τ)t, for τ ∈ G_{x}. (3)

From either description is easy to see that λ_{x} is a faithful character and hence a generator of the character group of G_{x}; in particular, ord(λ_{x}) = |G_{x}| = e_{x}. For future reference, let us note the formula

λ_{σx}(τ) = λ_{x}(σ^{−1}τσ) =: ^{σ}λ_{x}(τ), for τ ∈ G_{σx} = ^{σ}G_{x} = σG_{x}σ^{−1}, (4)

from which it follows that the induced characters

λ_{x̄,k} = λ_{G,x̄,k} = ind_{G_{x}}^{G}(λ_{x}^{k}) (5)

only depend on x̄ = π(x). Note that the latter characters satisfy the following restriction formula: if H ≤ G is any subgroup, then

(λ_{G,x̄,k})_{|H} = ∑_{x̃∈π̄^{−1}(x̄)} λ_{H,x̃,k}; (6)

this follows in view of Remark 1.1c) immediately from Mackey’s Subgroup Theorem (cf. [3], (10.13)) together with the fact that ^{τ}(λ_{x}^{k})_{|^{τ}G_{x}∩H} = λ_{H,τx,k}.

Although we are now ready to define the local invariants, it is best to defer the definition for a moment because for the application to the twisted diagonal quotient surfaces we require not only the local invariants themselves but also certain “twisted versions” of these which depend on a group automorphism α ∈ Aut(G). The reason for this is that we need to consider not only the “standard” action Φ : G × X → X of G on X given by Φ(σ, x) = σ · x, but also the twisted action

Φ_{α} : G × X → X

which is obtained by composing Φ with α; explicitly, Φ_{α}(σ, x) := Φ(α(σ), x) = α(σ) · x.

While this is essentially just a renaming of the group elements of G, and hence does not significantly alter the geometry of the action, some care must be taken with the action of subgroups, particularly if we consider several actions at the same time. We observe:

Remark 1.3 a) The quotient G\X = G\_{Φ_{α}}X is independent of α, as is the covering π = π_{Φ_{α}} : X → X̄ and hence also the set T = T_{Φ_{α},G} of fixed points of the action Φ_{α}. The stabilizer G_{x,α} := G_{Φ_{α},x} of a point x with respect to Φ_{α}, however, is G_{x,α} = α^{−1}(G_{x}) and hence is cyclic of order e_{Φ_{α},x} = e_{x}. Note that if τ ∈ G, then G_{α(τ)x,α} = ^{τ}G_{x,α} = τG_{x,α}τ^{−1}. Similarly, the local character of x ∈ X with respect to the twisted action is given by:

λ_{x,α} := λ_{Φ_{α},x} = λ_{x} ∘ α;

both are characters on G_{x,α} = α^{−1}(G_{x}).

In addition, if H ≤ G is any subgroup, then we have H\_{Φ_{α}}X = H^{α}\_{Φ}X and π_{Φ_{α},H} = π_{Φ,H^{α}}, π̄_{Φ_{α},H} = π̄_{Φ,H^{α}}, where H^{α} = α^{−1}(H).

b) If α = β_{σ} is the inner automorphism β_{σ}(τ) = σ^{−1}τσ, then we had seen above in Remark 1.1d) that X_{H^{α}} ≃ X_{H}. This, however, need not be true for an outer automorphism α, as the case of the Fermat curve X_{p} of exponent p shows: here G = ℤ/pℤ × ℤ/pℤ acts on X_{p}, and Aut(G) = Gl(ℤ/pℤ) acts transitively on the subgroups H ≤ G of order p, but three of the quotients X_{H} have genus 0, whereas the other p − 2 have genus \frac{1}{2}(p − 1).

We now come to the definition of the (twisted) local invariants which are fundamental for the study of diagonal quotient surfaces. Since these are closely linked to the coverings π_{H} attached to stabilizer subgroups H = G_{x,α}, we first introduce some convenient abbreviations for this map and other related objects.

Notation 1.4 a) If H = G_{x,α} = α^{−1}(G_{x}) is the stabilizer of a point x ∈ X with respect to the action Φ_{α}, then we write π_{x,α} = π_{H}, X_{x,α} = X_{H}, g_{x,α} = g_{H}, r_{x,α} = r_{H}, etc. for the associated subcover, the genus, the number of ramification points, and so on. Note that the isomorphism class of X_{x} depends only on x̄ = π(x), for if x′ = α(σ)x ∈ π^{−1}(x̄), then G_{x′,α} = ^{σ}G_{x,α} = ^{σ}H, and hence σ_{x,α} := σ_{H} : X_{x,α} → X_{^{σ}H} = X_{x′,α} defines an isomorphism (cf. Remark 1.1b)). We can thus write X_{x̄,α} = X_{x,α} and g_{x̄,α} = g_{x,α} in the sequel.

b) To define the local invariants, let x, y ∈ X be two points, and put G_{x,y,α} = G_{x}∩G_{y,α}, which is the stabilizer of x with respect to H = G_{y,α}. (Dually, α(G_{x,y,α}) is that of y with respect to H = G_{x,α^{−1}}.) Its order is denoted by

n_{α}(x, y) = |G_{x,y,α}| = e_{x}(G_{y,α}) = e_{y}(G_{x,α^{−1}}). (7)

We now define the integer q = q_{α}(x, y) by the condition:

(λ_{x})_{|K} = ((λ_{y,α})_{|K})^{q} and 0 ≤ q < n, (8)

where K = G_{x,y,α} and n = n_{α} = |K|. Such an integer exists because (λ_{x})_{|K} and (λ_{y,α})_{|K} are both faithful characters of K and hence are generators of the character group; thus, we also have that (n, q) = 1. We note that both n and q depend only on the image x̃ = π_{y,α}(x) of x in X_{y,α}, so we can write n_{α}(x̃, y) = n_{α}(x, y) and q_{α}(x̃, y) = q_{α}(x, y). For n this clear from its definition (7); for q we observe that if x′ = τx with τ ∈ H := G_{y,α}, then we have G_{x′,y,α} = H ∩ τG_{x}τ^{−1} = τ(H ∩ G_{x})τ^{−1} = K because H is abelian, and so by (4) we have (λ_{x′})_{|G_{x′,y,α}} = ^{τ}((λ_{x})_{|K}) = (λ_{x})_{|K} (using the commutativity of H again).

Finally, for x̄ ∈ X̄ and 1 ≤ ν ≤ e_{y}, we define s_{ν,α}(x̄, y) = |S_{ν,α}(x̄, y)| as the cardinality of the set

S_{ν,α}(x̄, y) = \{x̃ ∈ X_{x̄,y,α} : q_{α}(x̃, y) = \frac{ν}{(ν, e_{y})} and n_{α}(x̃, y) = \frac{e_{y}}{(ν, e_{y})}\} , (9)

where X_{x̄,y,α} = X_{H,x̄} = π̄_{y,α}^{−1}(x̄) denotes the fibre of π̄_{y,α} at x̄. We note that s_{ν,α}(x̄, y) depends only on the image ȳ = π(y) of y in X̄, so that we can write s_{ν,α}(x̄, ȳ) = s_{ν,α}(x̄, y). This follows from the fact that if y′ = α(σ)y ∈ π^{−1}(ȳ), then σ_{y,α}(S_{ν,α}(x̄, y)) = S_{ν,α}(x̄, y′), as is easy to check.

Remark 1.5 a) The local invariants s_{ν,α}(x̄, ȳ) may be viewed as a refinement of the ramification numbers introduced earlier. Indeed, by definition each x̃ ∈ S_{ν,α} has ramification index ex̃(π_{y,α}) = n_{α}(x̃, y) = \frac{e_{y}}{(ν,e_{y})}, so

r_{n,α}(x̄, ȳ) := ∑_{{ν=1 \atop (ν,e)= \frac{e}{n}}}^{e} s_{ν,α}(x̄, ȳ),

is just the number of points x̃ ∈ X_{x̄,y,α} = π̄_{y,α}^{−1}(x̄) with ramification degree ex̃(π_{y,α}) = n. For future reference, let us record here the formula

\frac{m}{ex̄} = ∑_{n|eȳ} r_{n}(x̄, ȳ)\frac{eȳ}{n} (10)

which follows from the fact that there are \frac{eȳ}{n} points on X above each x̃ ∈ S_{ν}(x̄, y) when n = (ν, eȳ), combined with the fact that there are \frac{m}{e_{x}} points in the fibre π^{−1}(x̄).

In a similar vein, if n > 1, then

r_{n,α}(ȳ) := ∑_{x̄∈X̄} r_{n,α}(x̄, ȳ) = ∑_{{ν=1 \atop (ν,e)= \frac{e}{n}}}^{e} s_{ν,α}(ȳ), where s_{ν,α}(ȳ) = ∑_{x̄∈X̄} s_{ν,α}(x̄, ȳ),

is the number of points x̃ ∈ X_{y,α} which are ramified of degree ex̃(π_{y,α}) = n, and hence

r_{ȳ,α} := r_{G_{y,α}} = ∑_{1<n|e} r_{n,α}(ȳ) = ∑_{ν=1}^{e−1} s_{ν,α}(ȳ), respectively t_{ȳ,α} := t_{G_{y,α}} = ∑_{1<n|e} \frac{e}{n}r_{n,α}(ȳ),

is the total number of ramified points (respectively, the number of fixed points) of π_{y,α} : X → X_{y,α}. Thus, by the Riemann–Hurwitz formula (1), the genus g_{ȳ,α} of X_{y,α} is given by

g_{ȳ,α} − 1 = \frac{g − 1}{eȳ} − \frac{1}{2} ∑_{1<n|eȳ} r_{n,α}(ȳ) (1 − \frac{1}{n}) = \frac{g − 1}{eȳ} − \frac{r_{ȳ,α}}{2} + \frac{t_{ȳ,α}}{2eȳ} . (11)

From this it follows that the following quantity 𝔾_{α} = 𝔾_{α}(X) = 𝔾_{α}(X, G), which will occur frequently in many expressions below, can be computed from the numbers r_{n,α}(ȳ) as follows:

𝔾_{α} := ḡ+(ḡ−1)g+\frac{1}{2} ∑_{ȳ∈X̄}(g−g_{ȳ,α}) = 2ḡ−1+\frac{(g − 1)^{2}}{m} +\frac{1}{4} ∑_{ȳ∈X̄} ∑_{1<n|eȳ} r_{n,α}(ȳ) (1 − \frac{1}{n}) .

(12)

b) From their definition it is evident that the local invariants s_{ν,α}(x̄, ȳ) are closely linked to the local characters λ_{x,α}. This relation is made more explicit by the following inner product formula which is valid for any x̄, ȳ ∈ X̄ and k, l ∈ ℤ when e = eȳ:

(λ_{x̄,k}, λ_{ȳ,α,l})_{G} = ∑_{{ν=1 \atop νk≡(ν,e)l (e)}}^{e} s_{ν,α}(x̄, ȳ) (13)

which follows easily from the restriction formula (6) by using Frobenius reciprocity.

c) Although the local invariants are defined asymmetrically, they satisfy a certain symmetry formula. This formula applies when ν \frac{ex̄}{eȳ} ∈ ℕ, which is no restriction since otherwise s_{ν}(x̄, ȳ) = 0 (cf. Proposition 1.7c) below). Then there is a unique integer ν^{∗} with 1 ≤ ν^{∗} ≤ ex̄ such that \frac{ex̄}{(ν^{∗},ex̄)} = \frac{eȳ}{(ν,eȳ)} and \frac{ν^{∗}}{(ν^{∗},ex̄)} \frac{ν}{(ν,eȳ)} ≡

1 mod (\frac{eȳ}{(ν,eȳ)}), and we have

s_{ν,α}(x̄, ȳ) = s_{ν^{∗},α^{−1}}(ȳ, x̄). (14)

To see this, note first that the map σ ↦ α(σ)^{−1} induces a bijection ᾱ_{x,y} : G_{y,α}\G/G_{x} →^{∼} G_{x,α^{−1}}\G/G_{y} and hence also via Remark 1.1c) a bijection ᾱ_{x,y} :

X_{x̄,y,α} = π̄_{y,α}^{−1}(x̄) →^{∼} X_{ȳ,x,α^{−1}} = π̄_{x,α^{−1}}^{−1}(ȳ). Then for x̃ ∈ X_{x̄,y,α} we have n_{α^{−1}}(α_{x,y}(x̃), x) =

n_{α}(x̃, y) and q_{α^{−1}}(α_{x,y}(x̃), x)q_{α}(x̃, y) ≡ 1 (mod n_{α}(x̃, y)), and so it follows that ᾱ_{x,y}(S_{ν}(x̄, y)) = S_{ν^{∗}}(ȳ, x), which proves (14).

d) The local invariants s_{ν,α}(x̄, ȳ) depend only on the image of α in the outer automorphism group Out(G) = Aut(G)/Inn(G). Indeed, if τ ∈ G, then we have n_{β_{τ} ∘α}(x, y) = n_{α}(x, τy) and q_{β_{τ} ∘α}(x, y) = q_{α}(x, τy), so τ_{y,α}(S_{ν,β_{τ} ∘α}(x̄, y)) = S_{ν,α}(x̄, τy), and hence it follows that s_{ν,β_{τ} ∘α}(x̄, ȳ) = s_{ν,α}(x̄, ȳ).

Since the local invariants s_{ν,α}(x̄, ȳ) will play such an important role in the sequel, it will be useful to have an efficient method for computing them. It turns out that this can be done by determining the following “normalizing sets”.

Notation 1.6 For σ, τ ∈ G put σ̄ = σ^{s/r}, τ̄ = τ^{t/r} where s = |σ|, t = |τ| and r = (s, t). Moreover, for any integer ν ≥ 1 put k = (ν, r) and define the relative normalizing sets

N_{ν}(σ, τ) = N_{k}(σ, τ) := {g ∈ G : gσ̄^{ν}g^{−1} ∈ ⟨τ̄^{ν}⟩} = {g ∈ G : g⟨σ̄^{ν}⟩g^{−1} = ⟨τ̄^{ν}⟩}
N_{ν}^{i}(σ, τ) = N_{k}^{i}(σ, τ) := {g ∈ G : gσ̄^{k}g^{−1} = τ̄^{ki}}
N_{ν}^{∗}(σ, τ) = N_{k}^{∗}(σ, τ) := N_{k}(σ, τ) \ ⋃_{d|k,d≠k} N_{d}(σ, τ)
N_{ν,i}^{∗}(σ, τ) = N_{k,i}^{∗}(σ, τ) := N_{ν}^{∗}(σ, τ) ∩ N_{ν}^{i}(σ, τ),

where i ∈ ℤ. Clearly N_{ν}^{i}(σ, τ) = N_{ν}^{j}(σ, τ) if i ≡ j (mod \frac{r}{k}) and N_{ν}^{i}(σ, τ) = ∅ if (i, \frac{r}{k}) ≠ 1, and similarly for N_{ν,i}^{∗}(σ, τ). Moreover, we have the following decompositions:

G = ⋃_{k|r}^{·} N_{ν}^{∗}(σ, τ), N_{ν}(σ, τ) = ⋃_{1≤i≤r}^{·} N_{ν}^{i}(σ, τ) and N_{ν}^{∗}(σ, τ) = ⋃_{1≤i≤r}^{·} N_{ν,i}^{∗}(σ, τ).

Note that H_{1} := ⟨σ⟩ and H_{2} := ⟨τ⟩) act on N_{ν}^{i} etc. by multiplication on the right and left, respectively, and hence all these sets are unions of (H_{2}, H_{1})-double cosets of G.

If τ = σ then we shall write N_{ν}(σ) = N_{ν}(σ, σ), N_{ν}^{∗}(σ) = N_{ν}^{∗}(σ, σ), and so on. Note that in this case N_{1}(σ) = N_{G}(⟨σ⟩) and N_{1}^{1}(σ) = C_{G}(σ) are just the usual normalizer and the centralizer of ⟨σ⟩, respectively.

These normalizing sets are connected to the invariants n_{α}(x, y), q_{α}(x, y) and s_{ν,α}(x̄, ȳ) in the following way.

Proposition 1.7 For x, y ∈ X, let σ_{x} and σ_{y} be generators of G_{x} = ⟨σ_{x}⟩ and of G_{y,α} = ⟨σ_{y}⟩. Let σ̄_{x}, σ̄_{y} and r = (e_{x}, e_{y}) be defined as above, and let c = c_{α}(σ_{x}, σ_{y}) be such that λ_{x}(σ̄_{x}) = λ_{y,α}(σ̄_{y})^{c}. Moreover, let ν ≥ 1 and put k = (ν, r). Then for g ∈ G we have:

a) g ∈ N_{ν}^{∗}(σ_{x}, σ_{y}) ⟺ n_{α}(gx, y) = \frac{r}{k} ⟺ G_{gx,y,α} = g⟨σ̄_{x}^{k}⟩g^{−1} = ⟨σ̄_{y}⟩.

b) If g ∈ N_{ν,i}^{∗}(σ_{x}, σ_{y}), then iq_{α}(gx, y) ≡ c (mod \frac{r}{k}). Thus, if ij ≡ c (mod \frac{r}{k}) and 1 ≤ j ≤ \frac{r}{k}, then g ∈ N_{ν,i}^{∗}(σ_{x}, σ_{y}) = N_{k,i}^{∗}(σ_{x}, σ_{y}) ⟺ π_{y,α}(gx) ∈ S_{jke_{y}/r,α}(x̄, y).

c) If ν′ := ν · \frac{r}{e_{y}} ∉ ℕ or if ν > e_{y}, then s_{ν,α}(x̄, ȳ) = 0. If ν′ ∈ ℕ and ν ≤ e_{y}, then we have a natural bijection

G_{y,α}\N_{ν′,cν̄^{∗}}^{∗}(σ_{x}, σ_{y})/G_{x} →^{∼} S_{ν,α}(x̄, y),

where ν̄^{∗} ∈ ℕ is such that ν̄^{∗} · \frac{ν}{(ν,e_{y})} ≡ 1 (mod \frac{e_{y}}{(ν,e_{y})}), and hence

s_{ν,α}(x̄, ȳ) = \frac{|N_{ν′,cν̄^{∗}}^{∗}(σ_{x}, σ_{y})|}{(ν, e_{y})e_{x}}.

Proof. a) The first equivalence clearly follows from the following assertion:

g ∈ N_{k}(σ_{x}, σ_{y}) ⟺ \frac{r}{k} | n_{α}(gx, y).

To prove this, note first that if g ∈ N_{k}(σ_{x}, σ_{y}) then g⟨σ̄_{x}^{k}⟩g^{−1} = ⟨σ̄_{y}^{k}⟩ ≤ gG_{x}g^{−1} ∩ G_{y,α} = G_{gx,y,α}, and so \frac{r}{k} = |⟨σ̄_{y}^{k}⟩| divides |G_{gx,y,α}| = n_{α}(gx, y). Conversely, if \frac{r}{k}|n_{α}(gx, y), then g⟨σ̄_{x}^{k}⟩g^{−1} and ⟨σ̄_{y}^{k}⟩ are subgroups of the cyclic group G_{gx,y,α} and hence must be equal, which means that g ∈ N_{k}(σ_{x}, σ_{y}). This proves the above assertion and hence the first equivalence. Moreover, the above proof shows that the second equivalence is also valid.

b) By the definition of q := q_{α}(gx, y) we have λ_{gx}(gσ̄_{x}^{k}g^{−1}) = λ_{y,α}(gσ̄_{x}^{k})^{q} = λ_{y,α}(σ̄_{y}^{ki})^{q} = λ_{y,α}(σ̄_{y})^{ikq}. On the other hand, we have λ_{gx}(gσ̄_{x}^{k}g^{−1}) = λ_{x}(σ̄_{x}^{k}) = λ_{y,α}(σ_{y})^{kc}, and so it follows that iq ≡ c (mod \frac{r}{k}), as asserted. Thus, if g ∈ N_{ν,i}^{∗}, then q = j and so π_{y,α}(gx) ∈ S_{ν̃,α} by the definition of Sν̃, if ν̃ = jke_{y}/r. This proves one implication, and the converse follows from this since the sets N_{k,i}^{∗} partition G and since i and j determine each other.

c) Suppose s_{ν,α}(x̄, y) ≠ 0, so there exists x̃ = π_{y,α}(gx) such that n_{α}(gx, y) = \frac{e_{y}}{(ν,e_{y})}. Since n_{α}(gx, y)|r = (e_{x}, e_{y}), we have \frac{e_{y}}{r} |(ν, e_{y})|ν, so ν′ ∈ ℕ. This proves the first statement. Moreover, we then have \frac{e_{y}}{(ν,e_{y})} = \frac{r}{(ν′,r)} and \frac{ν}{(ν,e_{y})} = \frac{ν′}{(ν′,r)}, and so by b) we see that the map f̄_{y,α} : g ↦ π_{y,α}(gx) of Remark 1.1c) induces the desired bijection. Thus, N_{ν′,cν̄^{∗}}^{∗}(σ_{x}, σ_{y}) is the disjoint union of precisely s_{ν,α}(x̄, ȳ) double cosets G_{y,α}gG_{x} with g ∈ N_{ν′,cν̄^{∗}}^{∗}(σ_{x}, σ_{y}). But each such double coset has |G_{y,α}gG_{x}| = \frac{e_{x}e_{y}}{n_{α}(gx,y)} = e_{x}(ν, e_{y}) elements, so |N_{ν′,cν̄^{∗}}^{∗}(σ_{x}, σ_{y})| = s_{ν,α}(x̄, ȳ)e_{x}(ν, e_{y}), which proves the desired formula.

1.2 The singular character h_{X}^{1}

The action of G on X induces an action on the singular cohomology space H^{1}(X, ℂ) as well as on the space H^{0}(X, Ω_{X}^{1}) of global holomorphic differential forms. The corresponding characters will be denoted by h^{1} = h_{X}^{1} and ω = ω_{X}, respectively. If we twist the action by α ∈ Aut(G), then the twisted action Φ_{α} induces a twisted action on H^{1}(X, ℂ) and H^{0}(X, Ω_{X}), whose corresponding characters are

h_{α}^{1} = h_{X,α}^{1} = h_{X}^{1} ∘ α and ω_{α} = ω_{X,α} = ω_{X} ∘ α. (15)

We now want to compute the inner product (h^{1}, h_{α}^{1})_{G} of the characters h^{1} and h_{α}^{1} since it will be interpreted below as a Betti number of the twisted diagonal surface Z_{α}.

Proposition 1.8 The character h_{α}^{1} of H^{1}(X, ℂ) with respect to the action Φ_{α} is given by

h_{α}^{1} = 2 · 1_{G} + (2ḡ − 2)reg_{G} + ∑_{x̄∈R}(reg_{G} − (1_{G_{x̄,α}})^{G}), (16)

where reg_{G} denotes the regular representation of G, R the ramification locus of π : X → X̄, and (1_{G_{x̄,α}})^{G} = ind_{G_{x,α}}^{G}(1_{G_{x,α}}) the induction to G of the trivial character 1_{G_{x,α}} of the stabilizer G_{x,α} = α^{−1}(G_{x}) ≤ G of any x ∈ π^{−1}(x̄). Thus, the inner product of h^{1} with its twist by α is

(17) (h^{1}, h_{α}^{1})_{G} = (h^{1}, h_{α^{−1}}^{1}) = 4𝔾_{α} = 4ḡ + 4(ḡ − 1)g + 2 ∑_{x̄∈X̄}(g − g_{x̄,α})
= 4ḡ(g + 1) + 2(r − 2)g − 2 ∑_{x̄∈R} g_{x̄,α}

Proof. By Serre [21], p. 106, h_{α}^{1} = 2 · 1_{G} + (2ḡ − 2)reg_{G} + ∑_{x̄∈X̄} ax̄, where ax̄ denotes the Artin character. Since we have tame ramification in characteristic 0, it follows from the definition of ax̄ that ax̄ = reg_{G} − (1_{G_{α,x}})^{G}, which proves (16).

The first equality in (17) is immediate, since for any two characters h, h′ of G we have (h, h′ ∘ α)_{G} = (h ∘ α^{−1}, h′)_{G} = (h′, h ∘ α^{−1})_{G}. Since the last two equalities are immediate from the definitions, it is enough to verify the second. For this, note first that for any subgroup H ≤ G we have

(h^{1}, (1_{H})^{G})_{G} = 2g_{H} (18)

because (h^{1}, (1_{H})^{G})_{G} = (h^{1}|_{H}, 1_{H})_{H} = dim(H^{1}(X)^{H}) = dim(H^{1}(X_{H})) = 2g_{H} by Frobenius reciprocity and the fact that H^{1}(X)^{H} = H^{1}(X_{H}); cf. Proposition 2.7 below. Thus, from (16) we obtain

(h^{1}, h_{α}^{1}) = 2(h^{1}, 1_{G}) + (2ḡ − 2)(h^{1}, reg_{G}) + ∑_{x̄∈X̄}((h^{1}, reg_{G}) − (h^{1}, (1_{G_{x̄,α}})^{G})
= 4ḡ + (2ḡ − 2)2g + ∑_{x̄∈X̄}(2g − 2g_{x̄,α})) = 4𝔾_{α},

which gives the desired second equation of (17).

1.3 The character ω_{X}

We now turn to the action of G on H^{0}(X, Ω_{X}^{1}). Its character ω was first determined by Chevalley and Weil [4] in terms of the local characters λ_{x} (cf. Notation 1.2). To state this theorem in a convenient form, we first introduce the following ramification characters w_{x}.

Notation 1.9 For x ∈ X we define the ramification character on G_{x,α} and on G by

w_{x,α} = w_{G,x,α} = ∑_{k=1}^{e_{x}−1} kλ_{x,α}^{k} and w_{x̄,α} = w_{G,x̄,α} = ∑_{x∈π^{−1}(x̄)}(w_{x,α})^{G}, (19)

respectively; here, x̄ = π(x). We note that the latter character can also be written in the form

w_{x̄,α} = \frac{m}{ex̄}w′_{x̄,α},

where w′_{x̄,α} = (w_{x,α})^{G} is independent of the choice of x ∈ π^{−1}(x̄) since for σ = α(τ) ∈ G we have λ_{σx,α} = ^{τ}λ_{x,α} = λ_{x,α} ∘ β_{τ}, with β_{τ} as in Remark 1.3b)).

The formula of Chevalley and Weil[4] may now be expressed as follows (cf. Kani[16], Corollary to Theorem 2):

Proposition 1.10 The character of H^{0}(X, Ω_{X}^{1}) as a G-module with respect to the action Φ_{α} is given by

ω_{α} = 1_{G} + (ḡ − 1)reg_{G} + \overline{w_{α}},

where w_{α} = w_{X,α} := \frac{1}{m} ∑_{x̄∈R} w_{x̄,α} = \frac{1}{m} ∑_{x∈X}(w_{x,α})^{G}.

Remark 1.11 The character ω_{α} is related to the singular character h_{α}^{1} by the formula

ω_{α} + \overline{ω_{α}} = h_{α}^{1};

this may be deduced either from Hodge-decomposition or from 1.10 and (16) since it is immediate that w_{x̄,α} + \overline{w_{x̄,α}} = max̄ = m(reg_{G} − (1_{G_{x,α}})^{G}).

Thus, if ω_{α} is real-valued, i.e. if ω_{α} = \overline{ω_{α}}, then ω_{α} is exactly one-half of the singular character h_{α}^{1}. In general, however, this is not the case; the difference is measured by the character μ_{α} := ω_{α} − \overline{ω_{α}} = w_{α} − \overline{w_{α}} as follows:

ω_{α} = \frac{1}{2}(h_{α}^{1} + μ_{α}). (20)

We observe that the correction term μ_{α} has a decomposition into local terms, for if we put μ_{ȳ,α} = w_{ȳ,α} − \overline{w_{ȳ,α}} then clearly

μ_{α} = \frac{1}{m} ∑_{x̄∈X̄} μ_{ȳ,α}. (21)

For future reference, let us note here that these local terms satisfy the following restriction property with respect to a subgroup H ≤ G:

(μ_{G,x̄})_{|H} = \frac{m}{n_{H}} ∑_{x̃∈π̄_{H}^{−1}(x̄)} μ_{H,x̃}; (22)

this follows easily from Mackey’s subgroup theorem; cf. Kani[16], Proposition 2.

We now want to derive a formula for the inner product (ω, \overline{ω_{α}}) which will later be interpreted as a geometric genus (cf. Proposition 3.1). It turns out that this inner product can be conveniently expressed in terms of certain Dedekind sums which we now introduce.

Notation 1.12 Let q and n be two relatively prime integers. Then the Dedekind sum (cf. [20], [14]) is defined by

𝕊(q, n) := ∑_{k=1}^{n−1} ((\frac{k}{n})) ((\frac{kq}{n} )) = ∑_{k=1}^{n−1} \frac{k}{n} ((\frac{kq}{n} )) = ∑_{k=1}^{n−1} \frac{k}{n} ⟨\frac{kq}{n} ⟩ − \frac{1}{4}(n − 1), (23)

where, as usual, ⟨x⟩ = x − [x] denotes the fractional part of x and ((x)) is the sawtooth function (i.e. ((x)) = ⟨x⟩ − \frac{1}{2}, if x ∉ ℤ and ((x)) = 0 if x ∈ ℤ). We now put, for y ∈ X and x̃ ∈ X_{y,α}:

𝕊_{α}(x̃, y) = 𝕊(q_{α}(x̃, y), n_{α}(x̃, y))
𝕊_{α}(x̄, y) = ∑_{x̃∈X_{x̄,y,α}} 𝕊_{α}(x̃, y) = ∑_{n=1}^{e_{y}} s_{n,α}(x̄, y)𝕊 (\frac{n}{(n, e_{y})}, \frac{e_{y}}{(n, e_{y})}) ,

where q_{α}(x̃, y), n_{α}(x̃, y) and s_{n,α}(x̄, y) = s_{n,α}(x̄, ȳ) are defined in Notation 1.4. From the last equality we see that 𝕊_{α}(x̄, y) only depends on ȳ = π(y), so we can write 𝕊_{α}(x̄, ȳ) = 𝕊_{α}(x̄, y). Finally, we define

𝕊_{α}(X) = ∑_{x̄,ȳ∈X̄} 𝕊_{α}(x̄, ȳ) = ∑_{n=1}^{e} s_{n,α}𝕊 (\frac{n}{(n, e)}, \frac{e}{(n, e)}) ,

where s_{n,α} = ∑_{x̄,ȳ∈X̄} s_{n,α}(x̄, ȳ) and e = lcm_{ȳ∈X̄}(eȳ).

Theorem 1.13 Let x̄, ȳ ∈ X̄. Then the inner product of the local correction terms is (μx̄, μ_{ȳ,α}) = 4m^{2}𝕊_{α}(x̄, ȳ), (24)

and hence the inner product of ω with its twisted contragredient character is

(ω_{X}, \overline{ω_{X,α}}) = \frac{1}{4}(h_{X}^{1}, h_{X,α}^{1}) − \frac{1}{4}(μ_{X}, μ_{X,α}) = 𝔾_{α}(X) − 𝕊_{α}(X); (25)

in particular, the inner product of the correction terms is (μ_{X}, μ_{X,α}) = 4𝕊_{α}(X).

Proof. Before deriving (24), let us see how (25) follows from it. For this, we first note that since clearly \overline{μ_{α}} = −μ_{α}, we have (h, μ_{α}) = 0 for any character h with \overline{h} = h. Thus, from (20) we obtain (ω, \overline{ω_{α}}) = \frac{1}{4}(h^{1} +μ, h_{α}^{1} −μ_{α}) = \frac{1}{4}((h^{1}, h_{α}^{1})−(μ, μ_{α})), which proves the first equality of (25). The second follows directly from (24) and the local sum formula (21). Finally, by substituting the formula (17) for (h^{1}, h_{α}^{1}), the second equality of (25) ensues.

It thus remains to prove (24). For this, we shall first derive the following refinement (26). To state it, fix x, y ∈ X, and put H = G_{y,α}, K = H_{x} = G_{x,y,α}, e = e_{y} = |H|, x̃ = π_{y,α}(x). Moreover, put μ_{y,α} = w_{y,α} − \overline{w_{y,α}}, so μ_{ȳ,α} = \frac{m}{e}ind_{H}^{G}(μ_{y,α}). Then we shall prove:

(w_{H,x̃}, μ_{y,α})_{H} = 2e^{2}𝕊_{α}(x̃, y). (26)

First note that (24) is an immediate consequence of this formula. Indeed, by Frobenius reciprocity and the restriction formula (22) we have

(27) (μx̄, μ_{ȳ,α})_{G} = 2(wx̄, μ_{ȳ,α})_{G} = \frac{2m}{e}(wx̄, ind_{H}^{G}(μ_{y,α}))_{G}
= \frac{2m}{e}(wx̄, (μ_{y,α})_{|H})_{H} = 2 (\frac{m}{e})^{2} ∑_{x̃∈π_{y,α}^{−1}(x̄)}(w_{H,x̃}, μ_{y,α}),

and so, in view of the definition of 𝕊(x̄, ȳ), we see that (24) follows directly from (26).

To prove (26), let us first compute (λ_{H,x̃,k}, w_{y,α}) for k ∈ ℤ, where λ_{H,x̃,k} is as defined in (5). Putting n = |K| = n_{α}(x̃, y), q = q_{α}(x̃, y), we claim that

(λ_{H,x̃,k}, w_{y,α})_{H} = e (⟨\frac{qk}{n} ⟩ + \frac{1}{2} (\frac{e}{n} − 1)) . (28)

To see this, put λ = (λ_{y,α})_{|K}. Then by the definition of q we have λ_{H,x}^{k} = λ^{qk}, so by Frobenius reciprocity we obtain

(λ_{H,x̃,k}, w_{y,α})_{H} = (λ_{H,x}^{k}, (w_{y,α})_{|K})_{K} = ∑_{j=1}^{e−1} j(λ^{qk}, λ^{j}) = ∑_{{j=1 \atop j≡qk(n)}}^{e−1} j

= ∑_{j=0}^{\frac{e}{n} −1} (n ⟨\frac{qk}{n} ⟩ + jn) = e (⟨\frac{qk}{n} ⟩ + \frac{1}{2} (\frac{e}{n} − 1)) ,

which proves (28). From this we obtain:

(λ_{H,x̃,k}, μ_{y,α})_{H} = e (2 ⟨\frac{qk}{n} ⟩ − 1) , if qk ≢ 0 (mod n) (29)

because \begin{aligned}(λ_{H,x̃,k}, μ_{y,α})_{H} & = & (λ_{H,x̃,k} − \overline{λ_{H,x̃,k}}, w_{y,α})_{H} = (λ_{H,x̃,k} − λ_{H,x̃,−k}, w_{y,α})_{H} \\ & = & e (⟨\frac{qk}{n} ⟩ + \frac{1}{2} (\frac{e}{n} − 1)) − (e (⟨\frac{−qk}{n} ⟩ + \frac{1}{2} (\frac{e}{n} − 1))) \\ & = & e (⟨\frac{qk}{n} ⟩ − ⟨\frac{−qk}{n} ⟩) = e (2 ⟨\frac{qk}{n} ⟩ − 1) , \end{aligned}

provided that qk ≢ 0 (mod n).

We can now finish the proof of (26) and hence the proof of the theorem. Indeed, using the definitions and formula (29), we obtain

(w_{H,x̃}, μ_{y,α}) = ∑_{k=1}^{n−1} k(\frac{e}{n}λ_{H,x̃,k}, μ_{y,α}) = \frac{e}{n} ∑_{k=1}^{n−1} ke (2 ⟨\frac{qk}{n} ⟩ − 1)

= 2e^{2} ∑_{k=1}^{n−1} \frac{k}{n} ((\frac{qk}{n} )) = 2e^{2}𝕊(q, n),

which proves (26).

Before leaving the topic of Dedekind sums, let us note that there is a curious relation between them and the lengths of finite continued fractions which will be useful later on. To be precise, these lengths are defined as follows.

Notation 1.14 Each fraction x = \frac{n}{q} > 1 has a unique continued fraction expansion of the form

\frac{n}{q} = [[c_{1}, c_{2}, …, c_{r}]] := c_{1} − \frac{1}{c_{2} − \frac{1}{... − \frac{1}{c_{r}}}} ,

where the coefficients c_{i} ≥ 2 are integers. Its length will be denoted by 𝕃(x) := r ≤ q.

Proposition 1.15 If 0 < q < n and (n, q) = 1, then

12𝕊(q, n) = 𝕃 (\frac{n}{n − q} ) − 𝕃 (\frac{n}{q}) + \frac{q}{n} + \frac{q^{∗}}{n} − 1 ≤ n − 1 − 𝕃 (\frac{n}{q}) ,

where q^{∗} is such that qq^{∗} ≡ 1 (mod n) and 0 < q^{∗} < n.

Proof. If q = 1 then q^{∗} = 1 and so 𝕃(\frac{n}{n−q}) + \frac{q}{n} + \frac{q^{∗}}{n} = n − 1 + \frac{2}{n} ≤ n, whereas if q ≥ 2 then 𝕃(\frac{n}{n−q}) + \frac{q}{n} + \frac{q^{∗}}{n} ≤ n − q + 1 + 1 ≤ n. This proves the asserted inequality.

To prove the identity, let \frac{n}{q} = [[c_{1}, …, c_{r}]] denote the continued fraction expansion of \frac{n}{q}. Since by Oda[19], Corollary 1.23 (p. 29) we have

∑_{i=1}^{r} c_{i} = 𝕃 (\frac{n}{n − q} ) + 2𝕃 (\frac{n}{q}) − 1,

it is enough to verify the formula (implicit in Hermann[11])

12𝕊(q, n) = ∑_{i=1}^{r} c_{i} − 3𝕃 (\frac{n}{q}) + \frac{q}{n} + \frac{q^{∗}}{n} . (30)

For this, we shall induct on r = 𝕃(\frac{n}{q}). If r = 1, then q = 1 and then 12𝕊(1, n) = n − 3 + \frac{2}{n} by [20], p. 5. Thus, assume r > 1 and write [[c_{2}, …, c_{r}]] = \frac{n_{1}}{q_{1}} with (n_{1}, q_{1}) = 1. Then \frac{n}{q} = c_{1} − \frac{q_{1}}{n_{1}} , and so n = c_{1}n_{1} − q_{1} and q = n_{1}. Thus 𝕊(n, q) = 𝕊(c_{1}n_{1} − q_{1}, n_{1}) = 𝕊(−q_{1}, n_{1}) = −𝕊(q_{1}, n_{1}), and so the Dedekind reciprocity formula yields:

12𝕊(q, n) = 12𝕊(q_{1}, n_{1}) + (−3 + \frac{n}{q} + \frac{1}{nq} + \frac{q}{n}).

Applying the induction hypothesis to \frac{n_{1}}{q_{1}} , we therefore obtain

12𝕊(q, n) = ∑_{i=1}^{r} c_{i} − 3r + \frac{q}{n} + \frac{nq_{1}^{∗} + 1}{nq},

where 0 < q_{1}^{∗} < n_{1} and q_{1}q_{1}^{∗} ≡ 1 (mod n_{1}).

It thus remains to show that nq_{1}^{∗} + 1 = qq^{∗}. For this, put q′ := (1 − q_{1}q_{1}^{∗})/n_{1} + c_{1}q_{1}^{∗} ∈ ℤ. Then qq′ = 1 − q_{1}q_{1}^{∗} + (n + q_{1})q_{1}^{∗} = 1 + nq_{1}^{∗}, so in particular qq′ ≡ 1 (mod n). Moreover, since 0 < qq′ = 1 + nq_{1}^{∗} < nn_{1} = nq, we have 0 < q′ < n. Thus q′ = q^{∗} and so nq_{1}^{∗} + 1 = qq^{∗}, as desired.

2 Diagonal quotient surfaces

2.1 Singularities

We now consider the product Y = X ×X of the curve X with itself. Since G acts on X, the product group G × G acts componentwise on the product surface Y. We shall be particularly interested in the induced action of the “graph subgroup”

Δ_{α} = {(σ, α(σ)) ∈ G × G : σ ∈ G}

defined by a group automorphism α ∈ Aut(G) of G. Since the map δ_{α} : σ ↦ (σ, α(σ)) defines a group isomorphism δ_{α} : G →^{∼} Δ_{α}, this action can also be considered as the action of G on Y given by (σ, (x, x′)) ↦ (σ · x, α(σ) · x′). The quotient

Z_{α} = G \ Y = Δ_{α} \ Y

of Y by this “twisted diagonal action” will be called a (twisted) diagonal quotient surface.

Let φ = φ_{α} : Y → Z_{α} = Δ_{α}\Y denote the quotient map. Since Δ_{α} ≤ G × G, we also have an induced quotient map ψ = ψ_{α} from Z_{α} to the product Ȳ = (G × G)\Y = X̄ × X̄ of X̄ with itself:

Y = X × X ⟶^{φ} Z_{α} = G \ Y ⟶^{ψ} Ȳ = X̄ × X̄.

Moreover, for i = 1, 2, we shall denote the composition of ψ with the i-th projection of Ȳ to X̄ by ψ_{i} = ψ_{α,i} = pr_{i} ∘ ψ_{α} : Z_{α} → X̄.

We begin our study of these surfaces by analyzing their singularities. For this, it is useful to first examine the fibres of the morphisms φ, ψ, and ψ_{i}.

Proposition 2.1 a) Let y = (x_{1}, x_{2}) ∈ Y be a point on the product surface Y. Then G_{y,α} := G_{x_{1},x_{2},α} = G_{x_{1}} ∩ α^{−1}(G_{x_{2}}) is the stabilizer of y with respect to the G-action on Y, so φ is ramified at y of order e_{y}(φ) = |G_{y,α}| = n_{α}(x_{1}, x_{2}), and hence the map f_{y} : τ ↦ τy = (τx_{1}, α(τ)x_{2}) induces a bijection

f_{y} : G_{y,α}\G →^{∼} φ^{−1}(φ(y)).

b) For i = 1, 2 and for each x̄ ∈ X̄, the reduced fibre C_{x̄,i} := ψ_{i}^{−1}(x̄)_{red} is a smooth irreducible curve on Z_{α}. More precisely, if we put α_{i} = α^{(−1)^{i}}, then for each x ∈ π^{−1}(x̄) we have isomorphisms

φ_{i} = φ_{x,i} : X_{x,α_{i}} = α_{i}^{−1}(G_{x})\X →^{∼} C_{x̄,i} = ψ_{i}^{−1}(x̄)_{red} ⊂ Z_{α}

which are characterized by the properties that φ_{1}(π_{x,α^{−1}}(x′)) = φ(x, x′) and φ_{2}(π_{x,α}(x′)) = φ(x′, x), if x′ ∈ X. Moreover, they satisfy the relations ψ(φ_{1}(x̃_{1})) = (x̄, π̄_{x,α^{−1}}(x̃_{1})) and ψ(φ_{2}(x̃_{2})) = (π̄_{x,α}(x̃_{2}), π(x)), for all x̃_{i} ∈ X_{x,α_{i}}.

c) The fibre of ψ at ȳ = ψφ(y) = (x̄_{1}, x̄_{2}) ∈ Ȳ is the intersection of the two curves C_{x̄_{i},i} and corresponds bijectively under φ_{x_{1},1} to the fibre X_{x̄_{2},x_{1},α^{−1}} of π̄_{x_{1},α^{−1}} at x̄_{2}, as well as under φ_{x_{2},2} to the fibre X_{x̄_{1},x_{2},α} of π̄_{x_{1},α} at x̄_{1}:

ψ^{−1}(ȳ) = C_{x̄_{1},1} ∩ C_{x̄_{2},2} = φ_{x_{1},1}(X_{x̄_{2},x_{1},α^{−1}}) = φ_{x_{2},2}(X_{x̄_{1},x_{2},α}).

In particular, the map τ ↦ φ(τx_{1}, x_{2}) induces a bijection

f̄_{y} : G_{x_{2},α}\G/G_{x_{1}} →^{∼} ψ^{−1}(ȳ)

between the double coset space G_{x_{2},α}\G/G_{x_{1}} and the fibre of ψ over ȳ.

Proof. a) The first assertion is clear, and the second follows in view of Remark 1.1b) since G acts transitively on the fibres of φ.

b) We first prove that C_{x̄,i} is smooth. For this, consider the curve C′ = φ^{−1}(C_{x̄,2}) = ∪_{x∈π^{−1}(x̄)}X × {x} on which G acts. Since the components of C′ are smooth and disjoint, C′ is locally normal and hence so is the quotient G\C′ (cf. Mumford[18], p. 5). Thus G\C′ is a normal, hence smooth curve. Since the natural morphism G\C′ → G\Y = Z_{α} is a closed immersion with image C_{x̄,2}, it follows that C_{x̄,2} is smooth. Similarly, one shows that C_{x̄,1} is smooth.

Next we construct the morphism φ_{2}. The map x′ ↦ φ(x′, x) is α^{−1}(G_{x})- equivariant because φ(τx′, x) = φ(x′, α(τ^{−1})x) = φ(x′, x) since α(τ^{−1}) ∈ G_{x}. Thus, the map factors over π_{x,α}, and so there a unique morphism φ_{2} : X_{x,α} → Z_{α} such that φ_{2}(π_{x,α}(x′)) = φ(x′, x). Moreover, if x̃_{2} = π_{x,α}(x_{2}) ∈ X_{x,α}, then ψ(φ_{2}(x̃_{2})) = ψφ(x_{2}, x) = (π(x_{2}), x̄) = (π̄_{x_{2},α}(x̃_{2}), x̄)), which yields the second formula.

In particular, we see that Im(φ_{2}) ⊂ ψ_{2}^{−1}(x̄)). To prove equality, let y′ = φ(x′_{1}, x′_{2}) ∈ ψ_{2}^{−1}(x̄). Then x′_{2} ∈ π^{−1}(x_{2}), so there is a τ ∈ G such that α(τ)x = x′_{2}. But then φ_{2}(τ^{−1}x′_{1}) = φ(τ^{−1}x′_{1}, x) = φ(x′_{1}, α(τ)x) = φ(x′_{1}, x′_{2}) = y′, so φ_{2} is surjective.

This shows that C_{x̄,2} = φ_{2}(X_{x,α}) is irreducible. Since we already know that C_{x̄,2} is smooth, it will therefore follow that φ_{2} is an isomorphism once we have verified that φ_{2} is injective.

To see this, let x̃′_{1} = π_{x,α}(x′_{1}), x̃′_{2} = π_{x,α}(x′_{2}) ∈ X_{x,α} be such that φ_{2}(x̃′_{1}) = φ_{2}(x̃′_{2}). Then we have φ(x′_{1}, x) = φ(x′_{2}, x), so there exits a τ ∈ G such that τx′_{1} = x′_{2} and α(τ)x = x. Thus τ ∈ α^{−1}(G_{x}) = G_{x,α}, so x̃′_{1} = π_{x,α}(x′_{1}) = π_{x,α}(τ^{−1}x′_{2}) = x̃′_{2}, and hence φ_{2} is injective.

The proof for φ_{1} is entirely analogous.

c) The first three assertions are obvious; in particular, we see that the bijection φ_{2} constructed above restricts to a bijection f̄_{2,y} : X_{x̄_{1},x_{2},α} → ψ^{−1}(ȳ). Next, consider the bijection f̄_{2} := f̄_{G_{x_{2},α}} : G_{x_{2},α}\G/G_{x_{1}} → X_{x̄_{1},x_{2},α} which is defined as in Remark 1.1c) by τ ↦ π_{x_{2},α}(τx_{2}). Then f̄_{2,y}(f̄_{2}(τ)) = f̄_{2,y}(π_{x_{2},α}(τx_{2})) = φ(τx, x_{2}) = f̄_{y}, which shows that our desired map f̄_{y} is well-defined and bijective.

We now investigate the singularities of Z_{α}. It turns out that they are all cyclic quotient singularities and hence are of the form A_{n,q} = G_{n}\ℂ^{2}; cf. BPV[1], p.

84. Explicitly, this means that the cyclic group G_{n} acts on ℂ^{2} via σ(z_{1}, z_{2}) = (χ_{1}(σ)z_{1}, χ_{2}(σ)z_{2}), where the “weights” χ_{i} : G_{n} → ℂ^{×} have order n and satisfy χ_{1} = χ_{2}^{q}.

Remark 2.2 There is a slight ambiguity concerning the type (n, q) of a quotient singularity A_{n,q} = G_{n}\ℂ^{2} in that it depends not only on the analytic isomorphism class of A_{n,q} but also on an ordering of the two weights χ_{i}. If we interchange the weights, then we obtain an (isomorphic) singularity of type (n, q^{∗}), with qq^{∗} ≡ 1 (mod n). However, this choice of an ordering is the only ambiguity, for two singularities of type A_{n,q} and A_{n′,q′} are analytically isomorphic if and only if n = n′ and q = q′ or qq′ ≡ 1 (mod n); cf. Hirzebruch [12], 3.4(18).

In our case, each quotient singularity z = φ(y) of Z_{α} is induced by the action of the stabilizer groups G_{y,α} on the tangent space T_{Y,y} = T_{X,x_{1}} ⊕T_{X,x_{2}} ≃ ℂ^{2} which comes equipped with a natural ordered basis (of eigenvectors), the ordering being induced by the ordering of the factors of Y = X × X. We thus have a natural ordering of the weights attached to this action, and hence (in this sense) a uniquely determined type (n, q).

Theorem 2.3 a) A point z = φ(y) = φ(x_{1}, x_{2}) ∈ Z_{α} is a singularity of Z_{α} if and only if e_{z}(φ) > 1 (cf. Proposition 2.1a)). If this is the case, then z is a cyclic quotient singularity of type A_{n,q}, where n = n(z) = n_{α}(x_{1}, x_{2}) = e_{z}(φ) and q = q(z) = q_{α}(x_{1}, x_{2}), where q_{α}(x_{1}, x_{2}) is as defined in 1.4.

b) For each ν ≥ 1, the map φ_{x_{2},2} of Proposition 2.1b) induces a bijection

f̄_{2,y,ν} : S_{ν,α}(π(x_{1}), x_{2}) → S_{ν,α}(ψ(z))

between the set S_{ν,α}(π(x_{1}), x_{2}) defined in 1.4 and the set S_{ν,α}(ψ(z)) of quotient singularities of fixed type (n, q) = (\frac{e}{(e,ν)}, \frac{ν}{(e,ν)}) in the fibre ψ^{−1}(ψ(z)), where e = e_{x_{2}}. In particular, the number of such singularities is s_{ν,α}(π(x_{1}), π(x_{2})).

c) In each fibre ψ_{2}^{−1}(x̄) there are precisely r_{x̄,α} singularities, where r_{x̄,α} is as defined in Remark 1.5. Similarly, in ψ_{1}^{−1}(x̄) there are r_{x̄,α^{−1}} singularities, and hence the total number of singularities of Z_{α} is

|S_{α}| = ∑_{x̄∈X̄} r_{x̄,α} = ∑_{x̄∈X̄} r_{x̄,α^{−1}}.

In particular, Z_{α} is smooth if and only if π_{x̄,α} : X → X_{x̄,α} is unramified for all x̄ ∈ X̄.

Proof. a) Let us factor φ as φ : Y →^{φ_{y}} Ỹ := G_{y,α}\Y →^{φ̃_{y}} Z_{α}, where G_{y,α} denotes the stabilizer of y with respect to G, and put ỹ = φ_{y}(y). Then φ̃_{y}^{∗} : O_{Z_{α},z}^{an} → O_{Ỹ ,ỹ}^{an} is unramified (cf. Bourbaki, Commutative Algebra, ch. V, §2.2, Prop. 4), hence local-etale (since Z_{α} is normal; cf. [5], Lemma I.1.5) and thus an isomorphism (cf. [5], Prop. 1.7). It follows that O_{Z_{α},z}^{an} is a quotient singularity with respect to the group G_{y,α}; in particular, if e_{z}(φ) = e_{y}(φ) = 1, then O_{Z_{α},z}^{an} ≃ O_{Y,y}^{an}, and so z is a smooth point of Z_{α}.

Now suppose that e_{z}(φ) > 1. Choose local coordinates t_{i} at x_{i} which linearize the action of G_{x_{i}} as in (3). Then (t_{1}, t_{2}) are local coordinates at y = (x_{1}, x_{2}) ∈ Y and we have

g^{∗}\binom{t_{1}}{t_{2}} = \binom{λ_{1}(g)t_{1}}{λ_{2}(g)t_{2}}, for g ∈ G_{y,α} = G_{x_{1}} ∩ α^{−1}(G_{x_{2}}),

where λ_{1} = λ_{x_{1}} and λ_{2} = λ_{x_{2},α}. Thus, the weights of the action are (λ_{1})_{|K} and (λ_{2})_{|K}, where K = G_{y,α}. Fix a generator τ of K, and write λ_{k}(τ) = e^{2πip_{k}/n}, k = 1, 2, where n = |K| = n_{α}(x_{1}, x_{2}) by Proposition 2.1a). We note that (p_{k}, n) = 1 since (λ_{k})_{|K} are both faithful characters. By definition (8) of q = q_{α}(x_{1}, x_{2}), we have λ_{1}(τ) = λ_{2}(τ)^{q}, so it follows that qp_{2} ≡ p_{1} (modn), and hence by Proposition 5.3 of BPV[1], p. 84, we have that z is a quotient singularity of type A_{n,q}.

b), c) These assertions follow immediately from Proposition 2.1b) and Remark 1.5a).

Remark 2.4 It is possible for G to have fixed points, yet that Z_{α} is smooth for a suitable twist α, for it can happen that all π_{x̄,α} : X → X_{x̄,α} are unramified, as the example of the Fermat curves shows (cf. Remark 1.3b)).

For computational purposes, it is useful to restate the above theorem in terms of the “normalizing sets” N_{k,i}^{∗}(σ_{1}, σ_{2}) introduced in Notation 1.6.

Corollary 2.5 Let x_{1}, x_{2} ∈ X, and let σ_{1} and σ_{2} be generators of G_{x_{1}} = ⟨σ_{1}⟩ and G_{x_{2},α} = ⟨σ_{2}⟩, respectively. Put r = (e_{1}, e_{2}) where e_{i} = e_{x_{i}}, and let c = c_{α}(σ_{1}, σ_{2}) be such that λ_{x_{1}}(σ_{1}^{e_{1}/r}) = λ_{x_{2},α}(σ_{2}^{e_{2}/r})^{c}. (31)

a) Let z ∈ ψ^{−1}(ψφ(x_{1}, x_{2})). Then z = φ(gx_{1}, x_{2}) for some g ∈ N_{k,i}^{∗}(σ_{1}, σ_{2}), where k and i satisfy 1 ≤ k|r, 1 ≤ i ≤ r and (i, \frac{r}{k}) = 1. Furthermore, k and i are uniquely determined by z.

b) The point z = φ(gx_{1}, x_{2}) is a singularity of Z_{α} if and only if k ≠ r. If this is the case, then its type (n, q) is uniquely determined by the relations

n = \frac{r}{k} and iq ≡ c (mod \frac{r}{k}). (32)

c) Let (k, i) and (n, q) be related by (32), and put ν = qe_{y}k/r. Then the map g ↦ φ(gx_{1}, x_{2}) induces a bijection

G_{x_{2},α}\N_{k,i}^{∗}(σ_{1}, σ_{2})/G_{x_{1}} →^{∼} S_{ν,α}(ψφ(x_{1}, x_{1})) (33)

between the indicated double coset space of normalizing elements and the set of singularities of type (n, q) which lie in the fibre ψ^{−1}(ψφ(x_{1}, x_{1})).

Proof. a) The first assertion is clear since the sets {N_{k,i}^{∗}(σ_{1}, σ_{2})}_{(k,i)∈S}, where S = \{(k, i) : 1 ≤ k|r, 1 ≤ i ≤ r, (i, \frac{r}{k}) = 1\} form a partition of G (cf. Notation 1.6). The uniqueness assertion follows immediately from part b).

b) By Theorem 2.3 we have n = n_{α}(gx_{1}, x_{2}) and q = q_{α}(gx_{1}, x_{2}). On the other hand, by Proposition 1.7a) we have n_{α}(gx_{1}, x_{2}) = \frac{r}{k} and by Proposition 1.7b) we see that q satisfies the congruence (32). This determines q uniquely since (i, \frac{r}{k}) = 1 and since 1 ≤ q ≤ n_{α} = \frac{r}{k} by the definition of q.

c) By composing the bijection of Proposition 1.7b), c) with that of Theorem 2.3b) we obtain a bijection between the two sets of (33). Now by (the proof of) Proposition 2.1c) we see that this composite bijection is the map induced by the map g ↦ φ(gx_{1}, x_{2}), and so the assertion follows.

As an application of (the proof of) Theorem 2.3, we can interpret the signature defect def_{y} (cf. Hirzebruch[13]) of a fixed point y ∈ Y of G in terms of an inner product of characters, and hence give a similar interpretation for the signature sign(Z_{α}) of Z_{α}:

Corollary 2.6 The signature defect of a point y = (x_{1}, x_{2}) ∈ Y with respect to G is

def_{y} = −4e_{y}(φ)𝕊_{α}(x̃_{1}, x_{2}) = −\frac{e_{y}(φ)}{e_{x_{2}}^{2}}(μ_{H,x̃_{1}}, μ_{x_{2},α})_{H}, (34)

where H = G_{x_{2},α}, x̃_{1} = π_{H}(x_{1}) and μ_{H,x̃_{1}}, μ_{x_{2},α} are as defined in 1.11. Thus we have

∑_{y′∈φ^{−1}ψ^{−1}(ȳ)} def_{y′} = −\frac{1}{m}(μx̄_{1}, μ_{x̄_{2},α})_{G} = 4m𝕊_{α}(x̄_{1}, x̄_{2}), (35)

where ȳ = ψ(φ(y)), and hence the signature of Z_{α} is

sign(Z_{α}) = −(μ_{X}, μ_{X,α}) = −4𝕊_{α}(X). (36)

Proof. Comparing the above computation (in the proof of 2.3) to that of Hirzebruch/Zagier [14], p. 179ff, we see from their formula (17) that their q = q_{α}(x_{1}, x_{2}) and p = n_{α}(x_{1}, x_{2}). (Moreover, their n = 1 and their m = 1.) Thus, by [14], formula (24) on p. 180, we obtain

def_{y} = −4p ∑_{k=1}^{p} ((\frac{k}{p} )) ((\frac{kq}{p} )) = −4p𝕊_{α}(x̃_{1}, x_{2}),

which yields the first equality of (34). On the other hand, by formula (26) we have

(μ_{H,x̃_{1}}, μ_{x_{2},α})_{H} = 2(w_{H,x̃_{1}}, μ_{x_{2},α})_{H} = 4e^{2}𝕊(x̃_{1}, x_{2}),

where e = e_{x_{2}}, from which the second equality of (34) follows.

To prove (35), first note that the second equality is just a restatement of (24). Next, since def_{y} depends only on z = φ(y), we obtain from (34) (and Prop. 2.1b)) and formula (27) that

∑_{y∈φ^{−1}ψ^{−1}(ȳ)} def_{y} = ∑_{z∈ψ^{−1}(ȳ)} \frac{m}{e_{z}(φ)}def_{z} = − ∑_{x̃∈X_{x̄_{1},x_{2},α}} \frac{m}{e^{2}}(μ_{H,x̃}, μ_{x_{2},α})_{H} = −\frac{1}{m}(μx̄_{1}, μ_{x̄_{2},α})_{G},

which proves the first equality of (35).

Finally, to prove (36), we shall apply the equivariant signature formula (cf. [13], formula (24) or [14], p. 181):

m · sign(Z_{α}) = sign(Y) + ∑_{y∈Y} def_{y}.

Since sign(Y) = 0 by [14], Th. 2.1.2, p. 26, we obtain from (35):

sign(Z_{α}) = \frac{1}{m} ∑_{ȳ∈Ȳ} ∑_{y∈(ψφ)^{−1}(ȳ)} def_{y} = −\frac{1}{m^{2}} ∑_{x̄_{1},x̄_{2}∈X̄}(μx̄_{1}, μ_{x̄_{2},α}) = −(μ_{X}, μ_{X,α}).

This proves the first equality of (36), and the second follows from (25).

2.2 The Betti and Hodge numbers

We now turn to calculate the Betti numbers b_{i}(Z_{α}) = h^{i}(Z_{α}, ℂ) as well as the Hodge numbers h^{0,i}(Z_{α}) = h^{i}(Z_{α}, O_{Z_{α}}) of Z_{α}. They can be computed from those of Y via

Proposition 2.7 For all q ≥ 0 we have

H^{q}(Z_{α}, ℂ) = H^{q}(Y, ℂ)^{Δ_{α}} and H^{q}(Z_{α}, O_{Z_{α}}) = H^{q}(Y, O_{Y})^{Δ_{α}}

where ( )^{G} denotes the G-invariant subspace.

Proof. Grothendieck [9], Corollary to Proposition 5.2.3 and Griffiths [7], formula (2.8).

Using this relation we find Proposition 2.8 The Betti numbers of Z_{α} are

b_{0}(Z_{α}) = b_{4}(Z_{α}) = 1, b_{1}(Z_{α}) = b_{3}(Z_{α}) = 4ḡ, b_{2}(Z_{α}) = 2 + (h^{1}, h_{α}^{1}) = 2 + 4𝔾_{α}.

In particular, the topological Euler characteristic of Z_{α} is

χ_{top}(Z_{α}) = 4(1 − 2ḡ + 𝔾_{α}) = 4(ḡ − 1)(g − 1) + 2 ∑_{x̄∈R}(g − g_{x̄,α}).

Proof. The first equation is obvious. For the other equations we consider H^{i}(Z_{α}, ℂ) as a G-module with corresponding character h^{i}(Z_{α}). Using the Künneth formula (and the fact that H^{i}(X, ℂ) = 0 for i > 2 = dim_{ℝ}(X)), we get

H^{1}(Y, ℂ) ≅ (H^{0}(X, ℂ) ⊗ H^{1}(X, ℂ)) ⊕ (H^{1}(X, ℂ) ⊗ H^{0}(X, ℂ)) ≅ H^{3}(Y, ℂ).

Observing the twisted action on the second factor of each summand, this implies h^{1}(Z_{α}) = h^{3}(Z_{α}) = h^{1} + h_{α}^{1}. Thus the preceding proposition gives b_{1}(Z_{α}) = b_{3}(Z_{α}) = (h^{1}(Z_{α}), 1_{G}) = (h^{1}, 1_{G}) + (h_{α}^{1}, 1_{G}) = 4ḡ by formula (18).

Similarly we get b_{2}(Z_{α}) = (h^{2}(Z_{α}), 1_{G}) = (1_{G} + h^{1} ⊗ h_{α}^{1} + 1_{G}, 1_{G}) = 2 + (h^{1} ⊗ h_{α}^{1}, 1_{G}) = 2 + (h^{1}, h_{α}^{1}) since h_{α}^{1} is real-valued (cf. equation (16)). Moreover, the inner product (h^{1}, h_{α}^{1}) was computed in Proposition 1.8, from which the last formula follows.

Proposition 2.9 The Hodge numbers h^{0,q} of Z_{α} are

h^{0,0} = h^{0}(O_{Z_{α}}) = 1, h^{0,1} = h^{1}(O_{Z_{α}}) = 2ḡ, h^{0,2} = h^{2}(O_{Z_{α}}) = (ω, \overline{ω_{α}}) = 𝔾_{α} − 𝕊_{α}.

In particular, the arithmetic genus of Z_{α} is given by

p_{a}(Z_{α}) = (ω, \overline{ω_{α}}) − 2ḡ = −1 + (ḡ − 1)(g − 1) + \frac{1}{2} ∑_{x̄∈R}(g − g_{x̄,α}) − 𝕊_{α}(X).

Proof. Let χ^{i} be the character of the G-module H^{i}(Y, O_{Y}) for i = 1, 2. From the Künneth formula and Serre duality on X we obtain

χ^{1} = \overline{ω} + \overline{ω_{α}} and χ^{2} = \overline{ω} · \overline{ω_{α}}.

Hence, applying the second part of Proposition 2.7, we obtain h^{1}(O_{Z_{α}}) = (χ^{1}, 1_{G}) = 2ḡ by formula (18), and h^{2}(O_{Z_{α}}) = (χ^{2}, 1_{G}) = (\overline{ω} · \overline{ω_{α}}, 1_{G}) = (ω, \overline{ω_{α}}). Finally, by substituting the expression (25) for (ω, \overline{ω_{α}}), the formula for p_{a} follows.

3 The desingularization of diagonal quotient surfaces

3.1 The geometric genus of Z̃_{α}

Since the diagonal quotient surface Z_{α} is almost always a singular surface (cf. Theorem 2.3c)), we shall mainly focus our attention on its (minimal) desingularization

σ : Z̃_{α} → Z_{α}.

In this subsection we shall to compute some of the numerical invariants of Z̃_{α} with the help of the group action on Y, whereas in the subsequent sections we shall take a closer look at the desingularization map σ to compute further invariants.

Proposition 3.1 The cohomology of the structure sheaf of Z̃_{α} is the same as that of Z_{α}. Therefore, the irregularity q, the geometric genus p_{g} and the arithmetic genus p_{a} of Z̃_{α} are given by

\begin{aligned}q(Z̃_{α}) & := & h^{1}(Z̃_{α}, OZ̃_{α}) & = & q(Z_{α}) & = & (h^{1}, 1_{G}) & = & 2ḡ, \\ p_{g}(Z̃_{α}) & := & h^{2}(Z̃_{α}, OZ̃_{α}) & = & p_{g}(Z_{α}) & = & (ω, \overline{ω_{α}}) & = & 𝔾_{α} − 𝕊_{α}, \\ p_{a}(Z̃_{α}) & := & p_{g}(Z̃_{α}) − q(Z̃_{α}) & = & p_{a}(Z_{α}) & = & (ω, \overline{ω_{α}}) − 2ḡ & = & 𝔾_{α} − 𝕊_{α} − 2ḡ. \end{aligned}

Proof. Since σ is birational and Z_{α} is normal, we have σ_{∗}OZ̃_{α} = O_{Z_{α}}. Moreover, we have R^{1}σ_{∗}OZ̃_{α} = 0 because by Proposition 2.3 all singularities of Z_{α} are cyclic quotient singularities and hence are rational; cf. Barth et al. [1] III Prop. (3.1). It thus follows from Hartshorne [10] Ex. III.8.1 that

H^{i}(Z̃_{α}, OZ̃_{α}) ≃ H^{i}(Z_{α}, σ_{∗}OZ̃_{α}) ≃ H^{i}(Z_{α}, O_{Z_{α}}) for all i ≥ 0.

This proves the first statement, and the others follow immediately from this in view of Proposition 2.9.

Corollary 3.2 If h_{α}^{1} = h^{1} but ω_{α} ≠ ω, then p_{g}(Z̃_{α}) > 0 if and only if g(X) > 0.

Proof. If (ω, \overline{ω_{α}}) = 0, then these characters have no common irreducible components, so h^{1} = ω + \overline{ω} = ω_{α} + \overline{ω_{α}} implies that ω_{α} = ω, which contradicts the assumption.

Corollary 3.3 If there is an automorphism \overline{α} of G such that ω\overline{α} = \overline{ω_{α}} then

p_{g}(Z̃_{α}) + p_{g}(Z̃\overline{α}) = \frac{1}{2}(h^{1}, h_{α}^{1}) = 2𝔾_{α},
p_{g}(Z̃_{α}) − p_{g}(Z̃\overline{α}) = −\frac{1}{2}(μ_{X}, μ_{X,α}) = \frac{1}{2}sign(Z_{α}) = −2𝕊_{α}.

Proof. By Theorem 3.1 we have

p_{g}(Z̃_{α}) ± p_{g}(Z̃\overline{α}) = (ω, \overline{ω_{α}}) ± (ω, \overline{ω\overline{α}}) = (ω, \overline{ω_{α}}) ± (ω, ω_{α}) = \frac{1}{2}(ω ± \overline{ω}, \overline{ω_{α}} ± ω_{α}),

where the last equality follows because (\overline{ω}, \overline{ω_{α}} ± ω_{α}) = (ω, ω_{α} ± \overline{ω_{α}}) = ±(ω, \overline{ω_{α}} ± ω_{α}). Since \overline{ω_{α}} + ω_{α} = h_{α}^{1} and \overline{ω_{α}} − ω_{α} = −μ_{α}, the first two assertions follow. The last equality is just a restatement of formula (36).

Remark 3.4 We shall see in [17] that the hypothesis of Corollary 3.3 is satisfied when X = X(N) is a modular curve. Thus, the above formula

p_{g}(Z̃_{α}) − p_{g}(Z̃\overline{α}) = \frac{1}{2}sign(Z_{α})

may be viewed as a generalization of Satz 3a) of Hermann[11], where this formula is proven in the special case that X = X(N).

3.2 The Euler characteristic of Z̃_{α}

To compute the Euler characteristic of Z̃_{α} we need to know that of the exceptional divisor E. We recall the following description of E given in BPV [1] III.5:

Proposition 3.5 If s ∈ Z_{α} is a singularity of type A_{n,q}, then

σ^{−1}(s) = ⋃_{j=1}^{r_{s}} C_{s,j} with C_{j} = C_{s,j} ≅ ^{ℙ}^{1}, C_{j}^{2} = −c_{s,j}, (C_{i}.C_{j}) = \{ \begin{matrix}1 & if |i − j| = 1 \\ 0 & if |i − j| > 1 \end{matrix} ,

where \frac{n}{q} = [[c_{s,1}, …, c_{s,r_{s}}]] is the continued fraction expansion of \frac{n}{q} of length 𝕃(\frac{n}{q}) = r_{s} as in Notation 1.14. Thus, if S_{α} denotes the set of singular points of Z_{α}, then the exceptional divisor of σ : Z̃_{α} → Z_{α} is

E = ∑_{s∈S_{α}} E_{s} where E_{s} = ∑_{j=1}^{r_{s}} C_{s,j},

and hence the number of irreducible components of E is

𝕃_{α}(X) := ∑_{x̄,ȳ∈X̄} 𝕃_{α}(x̄, ȳ) where 𝕃_{α}(x̄, ȳ) = ∑_{ν=1}^{eȳ−1} s_{ν}(x̄, ȳ)𝕃(\frac{eȳ}{ν}). (37)

Proposition 3.6 The topological Euler characteristic of Z̃_{α} is

χ_{top}(Z̃_{α}) = χ_{top}(Z_{α}) + #irreducible components of E = 4(1 − 2ḡ + 𝔾_{α}) + 𝕃_{α}.

Proof. For every closed subset A of a compact topological space T we have a long exact sequence

⋯ → H_{c}^{i}(T \ A, ℂ) → H^{i}(T, ℂ) → H^{i}(A, ℂ) → H_{c}^{i+1}(T \ A, ℂ) → · · · .

Applying this to (Z̃_{α}, E) and (Z_{α}, S_{α}) we get:

χ_{top}(Z̃_{α}) = χ_{top}(E) + χ_{top}^{c}(Z̃_{α} \ E),
χ_{top}(Z_{α}) = χ_{top}(S_{α}) + χ_{top}^{c}(Z_{α} \ S_{α}).

Since Z̃_{α} \ E ≅ Z_{α} \ S_{α} and S_{α} consists of isolated points this implies

χ_{top}(Z̃_{α}) = χ_{top}(Z_{α}) + χ_{top}(E) − |S_{α}| = χ_{top}(Z_{α}) + ∑_{s∈S_{α}}(χ_{top}(E_{s}) − 1).

Since

χ_{top}(E_{s}) = r_{s} + 1,

as is easily verified by using the Mayer-Vietoris sequence, the formula follows in view of Propositions 2.8 and 3.5.

3.3 The Betti, Hodge and Chern numbers of Z̃_{α}

By using results of the previous two subsections, we can now reap the fruits of our labour and compute the basic invariants of the surface Z̃_{α}.

Theorem 3.7 The Betti, Hodge and Chern numbers of the surface Z̃_{α} may all be expressed in terms of the three fundamental invariants of X, which were defined in 1.5, 1.12, and 3.5:

𝔾_{α} = ḡ + (ḡ − 1)g + \frac{1}{2} ∑_{x̄∈X̄}(g − g_{x̄,α})  = \frac{1}{4}(h^{1}, h_{α}^{1})
𝕊_{α} = ∑_{x̄∈X̄} ∑_{n=1}^{ex̄−1} s_{n,α}(x̄)𝕊 (\frac{n}{(n, ex̄)}, \frac{ex̄}{(n, ex̄)}) = \frac{1}{4}(h^{1}, h_{α}^{1}) − (ω, \overline{ω_{α}})
𝕃_{α} = ∑_{x̄∈X̄} ∑_{n=1}^{ex̄−1} s_{n,α}(x̄)𝕃 (\frac{ex̄}{n} ),

and hence also in terms of the numbers s_{n,α}(x̄). Explicitly, we have the following formulae:

h^{0,1}(Z̃_{α}) = q = \frac{1}{2}b_{1}(Z̃_{α}) = \frac{1}{2}b_{1}(Z_{α}) = 2ḡ
h^{0,2}(Z̃_{α}) = p_{g} = h^{0,2}(Z_{α})  = 𝔾_{α} − 𝕊_{α}
h^{1,1}(Z̃_{α}) = b_{2}(Z̃_{α}) − 2p_{g}(Z̃_{α})  = 2 + 2𝔾_{α} + 2𝕊_{α} + 𝕃_{α}
b_{2}(Z̃_{α}) = b_{2}(Z_{α}) + 𝕃_{α}  = 2 + 4𝔾_{α} + 𝕃_{α}.
sign(Z̃_{α}) = sign(Z_{α}) − 𝕃_{α}  = −4𝕊_{α} − 𝕃_{α}
c_{2}(Z̃_{α}) = χ_{top}(Z̃_{α}) = χ_{top}(Z_{α}) + 𝕃_{α} = 4(1 − 2ḡ + 𝔾_{α}) + 𝕃_{α}
c_{1}^{2}(Z̃_{α}) = 2c_{2}(Z̃_{α}) + 3sign(Z̃_{α})  = 8(1 − 2ḡ + 𝔾_{α}) − 𝕃_{α} − 12𝕊_{α}.

Proof. The first equality for 𝔾_{α} is just its definition (12), and the second is derived in Proposition 1.8. Similarly, the first equality for 𝕊_{α} is its definition (cf. Notation 1.12), whereas the second is formula (25), combined with (17). Moreover, the formula for 𝕃_{α} is just its definition (37).

The formulae for h^{0,1} and h^{0,2} are merely a restatement of Proposition 3.1, from which it also follows that b_{2}(Z̃_{α}) − b_{2}(Z_{α}) = χ_{top}(Z̃_{α}) − χ_{top}(Z_{α}) = 𝕃_{α}, the latter by Proposition 3.6. By substituting the values of b_{2}(Z_{α}) and χ_{top}(Z_{α}) which were obtained in Proposition 2.8, the indicated formulae for b_{2}(Z̃_{α}) and χ_{top}(Z̃_{α}) follow. (Recall that c_{2} = χ_{top} by the Gauss–Bonnet formula; cf. [8], p. 416.) From this the second identity for h^{1,1} follows immediately, whereas the first is just a consequence of the Hodge decomposition for H^{2}(Z̃_{α}, ℂ).

We can now compute the signature of Z̃_{α} via the formula

sign(Z̃_{α}) = b^{+} − b^{−} = (2p_{g} + 1) − (h^{1,1} − 1) = 2p_{g} − h^{1,1} + 2,

which follows from BPV[1], Theorem (IV.2.6), together with the fact that b^{+} + b^{−} = b_{2} = 2p_{g} + h^{1,1}. Substituting the above values for p_{g} and h^{1,1} yields sign(Z̃_{α}) = −4𝕊_{α} − 𝕃_{α} = sign(Z_{α}) − 𝕃_{α}, the latter by Corollary 2.6.

From this, the second formula for c_{1}^{2} follows immediately, whereas the first formula is the Index Theorem of Thom–Hirzebruch for Z̃_{α} (cf. [1], p. 18).

Corollary 3.8 For each n ∈ ℕ and x̄ ∈ X̄, let n^{∗} = n∗ ; x̄ be defined by the conditions 0 < n^{∗} ≤ ex̄, (n^{∗}, ex̄) = (n, ex̄) and \frac{nn^{∗}}{(n,ex̄)^{2}} ≡ 1 (mod \frac{ex̄}{(n,ex̄)}). Then we have

c_{1}^{2}(Z̃_{α}) = 8(1 − 2ḡ + 𝔾_{α}) − ∑_{x̄∈X̄} ∑_{n=1}^{ex̄−1} s_{n,α}(x̄) (𝕃 (\frac{ex̄}{ex̄ − n}) + \frac{n}{ex̄} + \frac{n^{∗}}{ex̄} − 1) (38)
≥ 8(1 − 2ḡ + 𝔾_{α}) − ∑_{x̄∈X̄} ∑_{1<k|ex̄} r_{k,α}(x̄)(k − 1).

In particular, if (as before) r = |R| denotes the number of ramified points in X̄, then

2c_{2}(Z̃_{α}) ≥ c_{1}^{2}(Z̃_{α}) ≥ (2g − 2)(r + 4(ḡ − 1)) + 2 ∑_{x̄∈R̄}(ex̄ − 2)(g_{x̄,α} − 1). (39)

Moreover, this lower bound is an equality if and only if ex̄ = 2 for all x̄ ∈ R for which π_{x̄,α} : X → X_{x̄,α} is ramified.

Proof. From Proposition 1.15 and the definitions it follows that

12𝕊_{α} + 𝕃_{α} = ∑_{x∈X̄} ∑_{n=1}^{ex̄−1} s_{n,α}(x̄) (𝕃 (\frac{ex̄}{ex̄ − n}) + \frac{n}{ex̄} + \frac{n^{∗}}{ex̄} − 1) , (40)

from which the formula follows in view of Theorem 3.7. Moreover, the stated inequality follows immediately from this and the inequality of Proposition 1.15, together with Remark 1.5.

To prove (39), note first that from (40) we see that 12𝕊_{α} + 𝕃_{α} ≥ 0. Thus, by Theorem 3.7 we have c_{1}^{2} ≤ 8(1 − 2ḡ + 𝔾_{α}) + 2𝕃_{α} = 2c_{2}, which proves the upper bound on c_{1}^{2}. To prove the lower bound, we use the estimate (k−1) ≤ \frac{ex̄}{k} (k−1) = ex̄(1 − \frac{1}{k}) and (11) to obtain

∑_{1<k|ex̄} r_{k,α}(x̄)(k − 1) ≤ ex̄ ∑_{1<k|ex̄} r_{k,α}(x̄)(1 − \frac{1}{k}) = (2g − 2) − ex̄(2g_{x̄,α} − 2).

Thus, since 1 − 2ḡ + 𝔾_{α} = (ḡ − 1)(g − 1) + \frac{1}{2} ∑_{x̄∈X̄}(g − g_{x̄,α}), we obtain

c_{1}^{2} ≥ 8(ḡ − 1)(g − 1) + ∑_{x̄∈X̄}(4(g − g_{x̄,α}) − (2g − 2) + ex̄(2g_{x̄,α} − 2)),

which yields the desired lower bound. Note that if this lower bound is an equality, then so is that of (38), which happens precisely when r_{n,α}(x̄) = 0 for n ≥ 3 (cf. the proof of Proposition 1.15). Moreover, from the above we see that we also require that (k − 1) = \frac{ex̄}{k} (k − 1) whenever r_{k,α}(x̄) ≠ 0, which means that ex̄ = 2 if π_{x̄,α} is ramified.

Remark 3.9 It follows from Remark 1.5d) that all the above invariants b_{i}(Z̃_{α}) etc. depend only on the class of α in the outer automorphism group Out(G). However, this may also be deduced more directly from the fact that if α′ = β_{τ} ∘α, then the map id × τ : Y → Y induces an isomorphism Z_{α} →^{∼} Z_{α′} which lifts to an isomorphism Z̃_{α} →^{∼} Z̃_{α′} of the desingularizations.

3.4 The canonical divisor

Although we have already computed the self-intersection number c_{1}^{2}(Z̃_{α}) = K_{α}^{2} of the canonical divisor class c_{1}(Z_{α}) = K_{α}, it will be often useful to have an explicit representative of the canonical divisor class at our disposal. Here we describe such a representative in two steps. First we give a conceptual description in Theorem 3.10, and then we work out explicitly the terms involved in Proposition 3.13.

Theorem 3.10 Let KȲ be a canonical divisor on Ȳ and let

D(ψ̃) := ∑_{x̄∈R} ∑_{i=1}^{2}(ψ̃^{∗}(C̄_{x̄,i}) − ψ̃^{−1}(C̄_{x̄,i})) + ∑_{s∈S_{α}} ∑_{j=1}^{r_{s}} C_{s,j}, (41)

where C̄_{x̄,1} = x̄ × Ȳ , C̄_{x̄,2} = Ȳ × x̄ and ψ̃^{−1}(C̄) = (ψ̃^{∗}(C))_{red} (viewed as a divisor on Z̃_{α}). Then

K_{α} = ψ̃^{∗}KȲ + D(ψ̃)

is a canonical divisor of Z̃_{α}.

Remark 3.11 a) The above theorem may be deduced from the general ramification formula (Iitaka [15], §5.6), viz.

K_{Z_{α}} ∼ ψ̃^{∗}(KȲ) + Rψ̃,

by showing that Rψ̃ = D(ψ̃). However, it is just as easy to prove Theorem 3.10 directly as it is to verify this identity.

b) Let C̃_{x̄,i} denote the proper transform of the curve C_{x̄,i} = ψ_{i}^{−1}(x̄)_{red} with respect to σ, which by Proposition 2.1b) is a smooth curve of genus g_{x̄,α_{i}}. Then by definition and 2.1, the divisor ψ̃^{−1}(C̄_{x̄,i}) consists of C̃_{x̄,i} together with all the exceptional divisors of σ which map to C̄_{x̄,i}, all with multiplicity one. However, the pullback ψ̃^{∗}(C̄_{x̄,i}) of C̄_{x̄,i} is more complicated to describe and will be determined in Proposition 3.13 below.

Proof of 3.10. Since the validity of the assertion does not depend on the choice of the canonical divisor KȲ, it is enough verify it for KȲ = div(η), where η is a two-form on Ȳ such that KȲ has no common component with ∑_{x̄∈R}(C̄_{x̄,1} + C̄_{x̄,2}). Put η̃ = ψ̃^{∗}(η) and

K_{α} := div(η̃) = ∑ n_{C}C.

The coefficients n_{C} can be computed locally as follows. For each z ∈ Z̃_{α} choose local coordinates (z_{1}, z_{2}) around ψ̃(z) of Ȳ. Thus η = fdz_{1} ∧ dz_{2} (locally). If there is a point z ∈ C at which ψ̃ is unramified, then (z_{1}, z_{2}) (or more precisely z_{i} ∘ ψ̃) are local coordinates around z and thus n_{C} is equal to the coefficient of ψ̃(C) in Kx̄.

Thus, suppose that all points of C are ramified with respect to ψ̃. Then either C = C̃_{x̄,i} with x̄ ∈ R or C = C_{s,j} is an exceptional curve.

For C = C̃_{x̄,1} with x̄ ∈ R, choose a point z ∈ C not lying over any (x̄_{1}, x̄_{2}) where x̄_{i} ∈ R. Then ψ̃ is ramified at z of index ex̄ and we can find local coordinates (z̃_{1}, z̃_{2}) around z such that z_{1} = z̃_{1}^{ex̄} and z_{2} = z̃_{2}. Therefore η̃ = fdz̃_{1}^{ex̄} ∧ dz̃_{2} = f(ex̄ −1)z̃_{1}^{ex̄−1}dz̃_{1} ∧dz̃_{2}. This implies n_{C} = ex̄ −1 because C_{x̄,1} is not a component of KȲ.

The same argument also gives the coefficients for C̃_{x̄,2}. Thus, any further components of K_{α} can only be exceptional ones, and hence

K_{α} = ψ̃^{∗}(KȲ) + ∑_{x̄∈X̄}(ex̄ − 1)(C̃_{x̄,1} + C̃_{x̄,2}) + D_{0},

where D_{0} ∈ D(S_{α}) := ⨁_{s,j} ℤC_{s,j} ≤ Div(Z̃_{α}) lies in the subgroup D(S_{α}) generated by the exceptional curves. On the other hand, from Proposition 2.1b) one easily deduces that

ψ̃^{∗}(C̄_{x̄,i}) = ex̄C̃_{x̄,i} + D_{i},

with D_{i} ∈ D(S_{α}), and so it follows that D := K_{α} − ψ̃^{∗}(KȲ) − D(ψ̃) ∈ D(S_{α}). Thus, to prove the assertion of the theorem (i.e. that D = 0), it is enough to verify that

(D.C_{s,j}) = 0, for all s ∈ S_{α}, 1 ≤ j ≤ r_{s},

because the intersection matrix (C_{s,j}.C_{s′,j′})_{s,s′,j,j′} is negative definite.

For this, fix s and j and write ψ(s) = (x̄_{1}, x̄_{2}). Then we have

((∑_{x̄,i} ψ̃^{−1}(C̄_{x̄,i}) − ∑_{s′,j′} C_{s′,j′} ) .C_{s,j}) = ((C̃_{x̄_{1},1} + C_{s,1} + … + C_{s,r_{s}} + C̃_{x̄_{2},2}) .C_{s,j}) = C_{s,j}^{2}+2,

where we have used Proposition 3.5 and also Lemma 3.12 below. On the other hand, by the adjunction formula we have

(K_{α}.C_{s,j}) = p_{a}(C_{s,j}) − C_{s,j}^{2} = −2 − C_{s,j}^{2},

whereas the projection formula yields

(ψ̃^{∗}(KȲ + ∑_{x̄,i} C̄_{x̄,i}).C_{s,j}) = ((KȲ + ∑_{x̄,i} C̄_{x̄,i}).ψ̃(C_{s,j})) = 0.

Combining these equations gives (D.C_{s,j}) = (−2 − C_{s,j}^{2}) − 0 + (2 + C_{s,j}^{2}) = 0, which proves the assertion of the theorem.

In the above proof we made use of the following crucial fact.

Lemma 3.12 Let s ∈ S_{α} and write ψ(s) = (x̄_{1}, x̄_{2}). Then for each x̄ ∈ X̄ and 1 ≤ j ≤ r_{s} we have

(C̃_{x̄,1}.C_{s,j}) = δ_{x̄x̄_{1}}δ_{j1} and (C̃_{x̄,2}.C_{s,j}) = δ_{x̄x̄_{2}}δ_{jr_{s}}.

Proof. Choose y = (x_{1}, x_{2}) ∈ Y such that φ(y) = s. Then by Proposition 2.1, φ(x_{1} × X) = C_{x̄_{1},1} and φ(X × x_{2}) = C_{x̄_{2},2}; recall that by definition C̃_{x̄_{i},i} is the proper transform of C_{x̄,i}.

As in the proof of Proposition 2.3b), let U be a small neighbourhood of y such that G_{y,α} acts linearly on U. Then, via the isomorphism U ≃ ℂ^{2} constructed there, the curves (x_{1} × X) ∩ U and X ∩ x_{2} ∩ U correspond to the coordinate axes z_{1} = 0 and z_{2} = 0, respectively. Thus, it follows from the explicit construction of the desingularization of the quotient singularity that the proper transform of the image of z_{1} = 0 (respectively, of z_{2} = 0) in G_{y,α}\ℂ^{2} meets the first component C_{s,1} (respectively, the last component C_{s,r_{s}}) transversally and none of the others, as is explained in detail in van der Geer [6], p. 42-3. Note, however, that the numbering used there is opposite of that used here. This proves the assertion if x̄ = x̄_{1} or x̄ = x̄_{2}. In the other cases we have s ∉ C_{x̄,1} or s ∉ C_{x̄,1}, so the curve C̃_{x̄,i} does not meet any C_{s,j}, and hence the assertion follows.

Although Theorem 3.10 is often already sufficient for many applications, it is useful to complement it by describing the the pullback of the divisors ψ̃^{∗}(C̄_{x̄,i}) explicitly.

Proposition 3.13 For each x̄ ∈ X̄, the pullback of the divisors C̄_{x̄,1} = x̄ × X̄ and C̄_{x̄,1} = X̄ × x̄ on Ȳ via ψ̃ = ψ ∘ σ is

ψ̃^{∗}(C̄_{x̄,i}) = ex̄C̃_{x̄,i} + ∑_{s∈S_{α}∩ψ_{i}^{−1}(x̄)} ∑_{j=1}^{r_{s}} a_{s,i,j}C_{s,j}, (42)

where i = 1, 2 and the coefficients a_{s,i,j} are determined from the continued fraction expansion of \frac{n_{s}}{q_{s}} = [[c_{s,1}, …, s_{s,r_{s}}]] (cf. Proposition 3.5) by the recursion relations

a_{s,i,j+1} = c_{s,j}a_{s,i,j} − a_{s,i,j−1}, 1 ≤ j ≤ r_{s}, (43)

together with the boundary conditions a_{s,1,0} = a_{s,2,r_{s}+1} = ex̄, a_{s,1,r_{s}+1} = a_{s,2,0} = 0. Thus we have

\begin{aligned}a_{s,1,1} & = & \frac{ex̄}{n_{s}} q_{s} & > & a_{s,1,2} & > & … & > & a_{s,1,r_{s}} & = & \frac{ex̄}{n_{s}} , \\ a_{s,2,1} & = & \frac{ex̄}{n_{s}} & < & a_{s,2,2} & < & … & < & a_{s,2,r_{s}} & = & \frac{ex̄}{n_{s}} q_{s}^{∗}, \end{aligned}

where q_{s}^{∗}q_{s} ≡ 1 (mod n_{s}) and 0 < q_{s}^{∗} < n_{s}. Furthermore, the divisor D(ψ̃) is given by

D(ψ̃) = ∑_{x̄∈X̄}(ex̄−1)(C̃_{x̄,1}+C̃_{x̄,2})+ ∑_{s∈S_{α}} ∑_{j=1}^{r_{s}}(a_{s,1,j}+a_{s,2,j}−1)C_{s,j} ≤ (ex̄−1)D(ψ̃)_{red}.

(44)

Proof. Clearly, ψ̃^{∗}(C̄_{x̄,i}) has a representation (42) with a_{s,i,j} ∈ ℕ. To determine the coefficients, fix i = 1, 2 and s ∈ S_{α} ∩ ψ_{i}^{−1}(x̄). The projection formula gives (ψ̃^{∗}(C̄_{x̄,i}).C_{s,j}) = (C̄_{x̄,i}.ψ̃_{∗}(C_{s,j}) = 0 for 1 ≤ j ≤ r_{s}. In view of Proposition 3.5 and Lemma 3.12, this leads to the system of recursion relations (43), together with the indicated boundary conditions, and these determine the a_{s,i,j} uniquely.

On the other hand, since \frac{n_{s}}{q_{s}} = [[c_{s,1}, …, c_{s,r_{s}}]], the sequence \{μ_{j}\} defined by μ_{0} = 0, μ_{1} = 1 and the recursion relation μ_{j+1} = c_{s,j}μ_{k} − μ_{j−1}, 1 ≤ j ≤ r_{s}, also satisfies μ_{r_{s}} = q_{s}^{∗} and μ_{r_{s}+1} = n_{s}, and similarly, the sequence {λ_{j}} defined by λ_{0} = n_{s}, λ_{1} = q_{s} and the same recursion relation also satisfies λ_{r_{s}} = 1, λ_{r_{s}+1} = 0; cf. BPV [1], p. 81. Thus, a_{s,1,j} = \frac{ex̄}{n_{s}} λ_{j}, a_{s,2,j} = \frac{ex̄}{n_{s}} μ_{j}, satisfy the recursion relation (43) and the boundary conditions. Furthermore, since by induction μ_{j+1} − μ_{j} ≥ … ≥ μ_{1} − μ_{0} = 1 and λ_{j} − λ_{j+1} ≥ … ≥ λ_{0} − λ_{1} = n_{s} − q_{s} ≥ 1, we see that the a_{s,i,j} satisfy the indicated values and inequalities.

The last formula/inequality follows immediately from the definition of D(ψ̃) and from the inequalities μ_{j} + λ_{j} ≤ n_{s}.

Remark 3.14 It is also possible to give determinantal expressions for the a_{s,i,j}’s. Indeed, if for integers d_{1}, …, d_{n} we put

[d_{1}, …, d_{n}] := det ( \begin{matrix}d_{1} & 1 & & 0 \\ 1 & ... & ... \\ & ... & ... & 1 \\ 0 & & 1 & d_{n} \end{matrix} ) and [ ] := 1,

then we have [−d_{1}, …, −d_{n}] = (−1)^{n}[d_{1}, …, d_{n}] and the Laplace expansion gives

[d_{1}, …, d_{n}] = d_{1}[d_{2}, …, d_{n}] − [d_{3}, …, d_{n}].

Comparing this recursion relation with (43) shows that we have

a_{s,1,j} = \frac{ex̄}{n_{s}} [c_{s,j+1}, …, c_{s,r_{s}}] and a_{s,2,j} = \frac{ex̄}{n_{s}} [c_{s,1}, …, c_{s,j−1}]. (45)

Corollary 3.15 For x̄ ∈ X̄ we have

(C̃_{x̄,1})^{2} = − ∑_{s∈S_{α}∩ψ_{1}^{−1}(x̄)} \frac{q_{s}}{n_{s}} = − ∑_{ȳ∈X̄} ∑_{ν=1}^{eȳ−1} s_{ν,α}(x̄, ȳ)\frac{ν}{eȳ},

(C̃_{x̄,2})^{2} = − ∑_{s∈S_{α}∩ψ_{2}^{−1}(x̄)} \frac{q_{s}^{∗}}{n_{s}} = − ∑_{ȳ∈X̄} ∑_{ν=1}^{ex̄−1} s_{ν,α}(ȳ, x̄)\frac{ν^{∗}}{ex̄} ,

where q_{s}^{∗} and ν^{∗} = ν∗ ; x̄ are as in 3.8. In particular, if α is such that α^{2} ∈ Inn(G), then C̃_{x̄,1}^{2} = C̃_{x̄,2}^{2}. Moreover, for any x̄, ȳ ∈ X̄ we have

(C̃_{x̄,1}.C̃_{ȳ,2}) = r_{1,α}(x̄, ȳ).

Proof. We first note that ψ̃_{∗}(C̃_{x̄,i}) = \frac{m}{ex̄} C̄_{x̄,i}. Thus, since (C̄_{x̄,i})^{2} = 0, the projection formula combined with (42) yields

0 = (ψ̃^{∗}(C̄_{x̄,i}).C̃_{x̄,i}) = ex̄(C̃_{x̄,i})^{2} + ∑_{s∈S_{α}∩ψ_{i}^{−1}(x̄)} ∑_{j=1}^{r_{s}} a_{s,i,j}(C_{s,j}.C̃_{x̄,i}),

so (C̃_{x̄,i})^{2} = − ∑_{s∈S_{α}∩ψ_{i}^{−1}(x̄)} ∑_{j=1}^{r_{s}}(a_{s,i,j}/ex̄)(C_{s,j}.C̃_{x̄,i}). Now for s ∈ ψ_{1}^{−1}(x̄) we have that ∑_{j=1}^{r_{s}}(a_{s,1,j}/ex̄)(C_{s,j}.C̃_{x̄,i}) = a_{s,1,1}/ex̄ = \frac{q_{s}}{n_{s}} by Lemma 3.12 and Proposition 3.13. This proves the first equality, and the second follows in view of 2.3b). The formulae for (C̃_{x̄,2})^{2} is proved analogously, and the identity C̃_{x̄,1}^{2} = C̃_{x̄,2}^{2} follows from the symmetry formula (14) and Remark 1.5d). A similar computation for (C̃_{x̄,1}.C̃_{ȳ,2}), using the fact that (C̄_{x̄,1}.C̄_{ȳ,2}) = 1, leads to

(C̃_{x̄,1}.C̃_{ȳ,2}) = \frac{m}{ex̄eȳ} − ∑_{s∈ψ^{−1}(x̄,ȳ)} \frac{1}{n_{s}} = \frac{m}{ex̄eȳ} − ∑_{1<d|eȳ} r_{d,α}(x̄, ȳ)\frac{1}{d},

which, by formula (10), yields the desired formula.

Remark 3.16 Using the above formulae it is possible to compute the the selfintersection number c_{1}^{2} = K_{α}^{2} once more; this leads to a second proof of Corollary 3.8, one that does not use Proposition 1.15.

4 Classification theorems

4.1 General results

In this section we want to examine how the twisted diagonal surfaces Z̃_{α} fit into the classification scheme of Enriques–Kodaira. Since it is rather difficult to determine the Kodaira dimension κ = κ(Z̃_{α}) in some cases, a complete classification does not seem to be possible; nevertheless, it is possible to obtain a partial classification which will be presented in this section. In the next sections we study criteria which are useful in handling some of the more subtle cases.

We begin by examining the effect of the genera g = g(X) and ḡ = g(X̄) on the Kodaira dimension.

Theorem 4.1 a) If g = 0 then Z̃_{α} is rational; i.e. κ = −∞.

b) If g = ḡ = 1, then Z̃_{α} ≃ Z_{α} is an abelian variety, so κ = 0.

c) If g = 1 and ḡ = 0, then Z̃_{α} is an elliptic surface with κ ≤ 0.

d) If g ≥ 2 and ḡ ≥ 1, then Z̃_{α} is a minimal surface of general type, i.e. κ = 2.

Proof. a) Since ℂ(Z̃_{α}) ⊂ ℂ(Y) and Y = ℙ^{1} × ℙ^{1} in this case, we see that Z̃_{α} is unirational and hence is rational by Castelnuovo’s theorem.

b) In this case Y = X × X is the product of two elliptic curves and hence is an abelian surface. Moreover, since g = ḡ, G acts without fixed points and hence consists of translations of X. Thus, Δ_{α} ≤ G × G consists of translations of Y and so Z_{α} = Δ_{α}\Y is again an abelian surface. In particular, Z_{α} is smooth and hence Z̃_{α} = Z_{α}.

c) Here Y is again an abelian surface, so κ(Z̃_{α}) ≤ κ(Y) = 0 (cf. [1], (I.7.4)). Moreover, almost all of the fibres of the fibration ψ̃_{i} : Z̃_{α} → X̄ ≃ ℙ^{1} are isomorphic to X (cf. Proposition 2.1b)), and hence are elliptic curves, so Z̃_{α} is an elliptic surface.

d) First note that p_{g} ≥ 1 because the canonical class formula 3.10 shows that we can find D ∼ K_{α}, D ≥ 0. (Here we use the fact that p_{g}(Ȳ) ≥ 1 because ḡ ≥ 1.) Thus, Z̃_{α} is not rational, so it is enough to show that K_{α}^{2} > 0, for then also the invariant K_{min}^{2} ≥ K_{α}^{2} of the associated minimal model Z_{min} is positive, and so Z_{min} and Z̃_{α} are of general type (cf. Beauville [2], X.1).

To show that K_{α}^{2} > 0, we use the inequality (39), which yields

K_{α}^{2} ≥ (2g − 2)(r + 4(ḡ − 1)) > 0

because g_{x̄,α} ≥ ḡ ≥ 1. Thus Z̃_{α} is a surface of general type.

Finally, we note that Z̃_{α} is minimal. Indeed, the only rational curves on Z̃_{α} are the curves C_{s,j}, for if C ≠ C_{s,j} were a rational curve on Z̃_{α}, then ψ̃_{i}(C) would be a rational curve covering X̄, which is impossible by Lüroth. But each C_{s,j} has self-intersection C_{s,j}^{2} ≤ −2, so is not a (−1)-curve. Thus, there are no (−1)-curves on Z̃_{α}, which means that Z̃_{α} is minimal.

In the above theorem, we determined the Kodaira dimension of all the surfaces except for those appearing in case c). This will be done next.

Theorem 4.2 If g = 1 and ḡ = 0, then Z̃_{α} is an elliptic surface with p_{g} ≤ 1 and K_{α}^{2} ≤ 0, and we have the following three possibilities:

a) If p_{g} = 0 and K_{α}^{2} < 0, then Z̃_{α} is rational (κ = −∞).

b) If p_{g} = 0 and K_{α}^{2} = 0, then Z̃_{α} is a minimal Enriques surface (κ = 0).

c) If p_{g} = 1, then Z̃_{α} is a minimal K3-surface (κ = 0).

Remark 4.3 All the cases listed in the theorem can actually occur. Cases a) and c) occur in the modular curve case, i.e. X = X(N) with N = 6; cf. [17]. Moreover, case b) also exists as we shall show now.

Example 4.4 An Enriques Surface. Let X be an elliptic curve with origin P_{0}, and let G = ⟨σ, τ⟩, where σ denotes the minus map σ(P) = −P, and τ is the translation map by some point P_{1} of order 2. Then G ≃ ℤ/2ℤ × ℤ/2ℤ, so there is an involution α ∈ Aut(G) such that α(σ) = τ. We claim that then p_{g}(Z̃_{α}) = 0, K_{α}^{2} = 0, so that Z̃_{α} is an Enriques surface by Theorem 4.2.

To see this, note first that ḡ = 0 since σ has a fixed point. Furthermore, we have r = 4 and ex̄ = 2 for all x̄ ∈ R. Now G_{x} = ⟨σ⟩ for x ∈ π^{−1}(x̄), for two (say x̄ = x̄_{1}, x̄_{2}) of the four points x̄ ∈ R, and hence g_{x̄,α} = 1 for these by our choice of α. We thus have r_{x̄_{i},α} = 0 for i = 1, 2 and r_{x̄_{i},α} = 4 for i = 3, 4. It thus follows from the definitions that 𝔾_{α} = 𝕊_{α} = 0 and 𝕃_{α} = 8, and hence by Theorem 3.7 we obtain p_{g} = 𝔾_{α} − 𝕊_{α} = 0 and K_{α}^{2} = 8(1 + 𝔾_{α}) − 𝕃_{α} + 12𝕊_{α} = 0, which proves the claim.

Note that if we take α = id, then 𝔾_{id} = 1, 𝕊_{id} = 0 and 𝕃_{α} = 16, so p_{g}(Z̃_{id}) = 1, K_{id}^{2} = 0 and hence Z̃_{α} is a K3-surface in this case.

The proof of Theorem 4.2 is based on the following facts.

Proposition 4.5 Suppose g ≥ 1 and ḡ = 0. Then:

a) If H^{0}(Z̃_{α}, −K_{α}) ≠ 0, then g = 1 and either K_{α} = 0 or K_{α}^{2} < 0.

b) The second plurigenus P_{2} = P_{2}(Z̃_{α}) satisfies the inequality

P_{2} ≥ K_{α}^{2} + 1 + p_{g},

except when g = 1 and K_{α} = 0, i.e. except when Z̃_{α} is a K3-surface.

Before we prove this Proposition, let us note the following important consequence.

Corollary 4.6 If g ≥ 1 and K_{α}^{2} > 0 then Z̃_{α} is of general type (κ = 2).

Proof. By Theorem 4.1b), d) we may assume that ḡ = 0. Then, since K_{α} ≠ 0, we have by Proposition 4.5b) that P_{2} ≥ K^{2} + 1 + p_{g} > 0, so Z̃_{α} cannot be rational. Thus, since K_{α}^{2} > 0, it follows that Z̃_{α} is of general type (cf. proof of Theorem 4.1).

The proof of Proposition 4.5 depends on the following simple fact.

Lemma 4.7 Suppose that ḡ = 0 and that D > 0 is an effective (non-zero) divisor on Z̃_{α} with (D.C̃_{x̄,i}) ≤ 0 for some x̄ ∉ R and for i = 1, 2. Then D is a linear combination of the exceptional curves C_{s,j} and hence D^{2} < 0 and (D.C̃_{x̄,i}) = 0 for x̄ ∉ R and i = 1, 2.

Proof. Since D̄ := ψ̃_{∗}(D) is an effective divisor on Ȳ = ℙ^{1} × ℙ^{1} with (D̄.C̄_{x̄,i}) = (D.C̃_{x̄,i}) ≤ 0, we must have D̄ = 0. Thus D consists entirely of curves blown down under ψ̃. This proves the first statement, and the second follows since the intersection pairing is negative definite on the space spanned by the C_{s,j}’s. The last is clear by the projection formula since D̄ = 0. Proof of Proposition 4.5. a) By hypothesis, there is an effective divisor D ≥ 0 such that D ∼ −K_{α}. Let x̄ ∉ R. Then C̃_{x̄,i}^{2} = 0, so by the adjunction formula (D.C̃_{x̄,i}) = −(K_{α}.C̃_{x̄,i}) = −(2g−2). From Lemma 4.7 we therefore see that g ≤ 1, hence g = 1 and that moreover D^{2} < 0 if D ≠ 0. Since D ∼ −K, the assertion follows.

b) We may assume that K_{α} ≥ 0, for otherwise the statement follows from the trivial inequality P_{2} ≥ p_{g}. Then by part a) we have either g = 1 and K_{α} = 0 or h^{0}(−K_{α}) = 0. In the former case Z̃_{α} is clearly (by definition) a K3-surface; recall that q = ḡ = 0. In the latter case we have by Riemann–Roch (and Serre duality) on Z̃_{α}:

P_{2} = h^{0}(2K_{α}) = h^{0}(2K_{α})+h^{0}(−K_{α}) ≥ \frac{1}{2}(2K_{α}).(2K_{α}−K_{α})+1+p_{a} = K_{α}^{2}+1+p_{g},

which proves the desired inequality.

Proof of Theorem 4.2. The hypotheses imply that ψ̃_{i} : Z̃_{α} → X̄ is an elliptic fibring; cf. 2.1b). The second assertion follows from the inequality p_{g}(Z̃_{α}) ≤ p_{g}(Y) = 1, and the third follows from Corollary 4.6 since we already know that κ ≤ 0 by Theorem 4.1c).

a) Suppose not; then by Castelnuovo’s criterion we must have P_{2} ≥ 1 and hence κ = 0. Let D ∈ |2K_{α}| ≠ ∅. Then, as in the proof of 4.5, we have (D.C̃_{x̄,i}) = 2(2g − 2) = 0, for all x̄ ∉ R, and so by Lemma 4.7 it follows that all components of D are of the form C_{s,j}; in particular, D does not contain any (−1)-curves. But this means that Z̃_{α} is minimal, for any (−1)-curve is contained in some D ∈ |2K_{α}| because P_{2} ≥ 1. But there is no minimal surface with P_{2} ≥ 1, K^{2} < 0 and κ = 0 (cf. Beauville[2], VIII.2); contradiction. It thus follows that Z̃_{α} must be rational.

b) Since p_{g} = 0, K_{α} ≠ 0, so from Proposition 4.5 we obtain P_{2} ≥ 0+1+0 = 1. This shows that Z̃_{α} cannot be rational, and hence κ = 0. Thus, since p_{g} = 0 and q = 0, this means that Z̃_{α} is an Enriques surface; cf. Beauville[2], VIII.2.

c) If K_{α} ≠ 0, then Proposition 4.5 yields P_{2} ≥ 0+1+1 = 2, which is impossible for a surface with κ = 0. Thus K_{α} = 0 and hence Z̃_{α} is a K3-surface.

Since Theorems 4.1 and 4.2 completely classify the cases that g ≤ 1 or that ḡ ≥ 1, we are left with the case that g ≥ 2 and ḡ = 0. Here, however, it seems to be impossible to give a classification based on the behaviour of g alone. Nevertheless, one might expect that Z̃_{α} is of general type if m = |G| is small relative to g. Since the latter condition is true if r = |R| is large, the following theorem substantiates this expectation to some extent.

Theorem 4.8 Z̃_{α} is of general type if any one of the following conditions holds:

a) r ≥ 8;

b) r ≥ 6 and G is not cyclic;

c) r ≥ 4, g ≥ 2 and g_{x̄,α} ≥ 1, for all x̄ ∈ R.

Proof. In all three cases we shall apply Corollary 4.6 and hence have to verify that K_{α}^{2} > 0 in these cases. Note that the hypotheses imply that g ≥ 2 because 2g − 2 ≥ m(−2 + \frac{r}{2}) > 0 if r ≥ 5.

a) By Corollary 3.8, specifically by inequality (39), we have

K_{α}^{2} ≥ (2g − 2)(r − 4) + 2 ∑_{x̄∈R}(ex̄ − 2)(g_{x,α} − 1) ≥ (2g − 2)(r − 4) − 2 ∑_{x̄∈R}(ex̄ − 2).

Let r_{n} = #{x̄ ∈ R : ex̄ = n}. Then ∑_{x̄∈R}(ex̄ − 2) = ∑_{1<d|m} r_{d}(d − 2), so

∑_{x̄∈R}(ex̄ − 2) ≤ ∑_{1<d|m} r_{d} \frac{m}{d}(d − 2) = − ∑_{1<d|m} mr_{d} + 2 ∑_{1<d|m} mr_{d} (1 − \frac{1}{d})
= −rm + 2[(2g − 2) − m(2ḡ − 2)].

Since we may assume that ḡ = 0 (cf. Theorem 4.1d)), we obtain

K_{α}^{2} ≥ (2g − 2)(r − 8) + 2m(r − 4) > 0.

b) If G is not cyclic, then r_{m} = 0, so we have r_{d} ≤ \frac{m}{2d}r_{d} for all d|m, and hence in this case the argument of a) leads to the inequality

K_{α}^{2} ≥ (2g − 2)(r − 6) + m(r − 4) > 0.

c) From Corollary 3.8 we see directly that K_{α}^{2} ≥ 0. Suppose K_{α}^{2} = 0. Then (39) is an equality, so r = 4 and also ex̄ = 2 if π_{x̄,α} is ramified by the last assertion of Corollary 3.8. Now if ex̄ = 2 for all x̄ ∈ R, then 2g − 2 = m(−2 + 4 · \frac{1}{2}) = 0, contradiction. Thus, there is at least one x̄ ∈ R with ex̄ ≥ 3. Then π_{x̄,α} must be unramified, so g_{x̄,α} > 1. But then (39) yields K_{α}^{2} > 0, as claimed.

Remark 4.9 In Theorem 4.8, the main criterion employed for proving that Z_{α} is of general type was that K_{α}^{2} > 0; cf. Corollary 4.6. Now if Z̃_{α} were minimal, then this condition would be also necessary; however, the surfaces Z̃_{α} are rarely minimal, as specific examples show; cf. [17]. We also have the following example of a more general nature:

The surface Z̃_{α} is never minimal if α = id, r = 3 and ḡ = 0.

To see this, we shall show that Γ̃ := the proper transform of (the image of) the diagonal Γ = φ({(x, x) ∈ Y : x ∈ X}) is an exceptional (−1)-curve.

Since the diagonal on Y is invariant under the untwisted diagonal action, we have that Γ is isomorphic to G\X and hence is smooth of genus ḡ, and the same is true for Γ̃. Furthermore, since ψ̃_{∗}Γ̃ = Γ̄ is the diagonal of Ȳ, the projection formula and 3.13 imply

1 = (C̄_{x̄,i}.Γ̄) = ex̄(C̃_{x̄,i}.Γ̃) + ∑_{s∈S_{α}} ∑_{j=1}^{r_{s}} a_{s,i,j}(C_{s,j}.Γ̃).

Since all these terms are non-negative and ex̄ ≥ 2 if x̄ ∈ R, it follows that

(Γ̃.C̃_{x̄,i}) = 0, ∀x̄ ∈ R and (C_{s,j}.Γ̃) = δ_{s,sx̄}, ∀s, j,

where sx̄ = φ(x, x) for x ∈ π^{−1}(x̄). It thus follows from (44) that (D(ψ̃).Γ̃) = r because a_{sx̄,i,1} = r_{sx̄} = 1. Thus, the canonical class formula 3.10 gives (K_{α}.Γ̃) = (KȲ .Γ̄) + (D(ψ̃).Γ̃) = 4ḡ − 4 + r, and so the adjunction formula yields Γ̃^{2} = 2 − 2ḡ − r. Thus, Γ̃ is a (−1)-curve if r = 3 and ḡ = 0.

4.2 Criteria for general type

In order to be able to treat some of the cases for which the criterion of Corollary 4.6 may fail (cf. Remark 4.9), we introduce here the following invariant d(S, C) which often helps to resolve the issue.

Notation 4.10 Let C be a curve on a smooth surface S. We put

d(S, C) := 2 (p_{g}(S) + q(S) − p_{a}(C)) + (C.C) = 2(p_{g}(S) + q(S) − 1) − (K_{S}.C),

where the last equality follows from the adjunction formula.

This invariant satisfies the following properties.

Proposition 4.11 a) If f : S → S′ is a birational morphism of smooth surfaces and C is an irreducible curve on S such that f(C) is not blown down to a point, then

d(S, C) ≤ d(S′, f(S)).

b) Let f : S → B be a (minimal) elliptic fibration with multiple fibres m_{1}F_{1}, …, m_{t}F_{t} where t ≥ 0. If g(B) = q(S), then for any irreducible curve C on S we have

d(S, C) = 2(p_{g} + q − 1) − deg_{f}(C) (p_{g} + q − 1 + ∑_{i=1}^{t} (1 − \frac{1}{m_{i}} )) ,

where deg_{f}(C) = (C.f^{∗}(P)), for any P ∈ B. In particular, if C is neither a section of f nor a component of a fibre of f, then d(S, C) ≤ 0, and even d(S, C) < 0 if f has multiple fibres.

Proof. a) By the factorization lemma and induction, it is enough to verify this when f is the blow-down map of a (−1)-curve E. Then K_{S} = f^{∗}K_{S′} + E, and hence we have (K_{S}.C) = (K_{S′}.f_{∗}(C))+(E.C). Thus, since p_{g} and q are birational invariants, d(S, C) − d(S′, f_{∗}(C)) = −(E.C) ≤ 0 because C ≠ E.

b) By Kodaira’s formula (cf. BPV[1], V.12.3) we have K_{S} = f^{∗}(D)+∑_{i=1}^{t}(m_{i}− 1)F_{i}, where deg(D) = χ(O_{S}) − 2χ(O_{B}) = p_{g} + q − 1 (since g(B) = q(S)). Thus (K_{S}.C) = (C.f^{∗}(P))(p_{g} + q − 1 + ∑(m_{i} − 1)\frac{1}{m_{i}}), and so the formula follows. The second assertion is clear, since the we have deg_{f}(C) ≥ 2 in this case.

Theorem 4.12 Let S be a smooth regular surface with κ(S) ≥ 1. Then S is of general type if any one of the following conditions holds:

a) There is an irreducible curve C on S of genus p_{a}(C) ≥ 2 and with d(S, C) ≥ 1.

b) There exist two intersecting elliptic curves E_{1} and E_{2} on S with d(S, E_{i}) ≥ 1.

c) There exist t ≥ 1 distinct elliptic curves E_{1}, …, E_{t} on S with d(S, E_{i}) ≥ 1 for 1 ≤ i ≤ t such that

K_{S}^{2} > ∑_{i=1}^{t} E_{i}^{2}. (46)

Proof. Let S̄ denote the minimal model of S and σ : S → S̄ the associated birational map. Suppose that S, hence S̄, is not of general type. Then κ = 1, so there is an elliptic fibration π̄ : S̄ → B to some smooth curve B (cf. Beauville[2], IX.2) of genus g(B) = q(S̄) or g(B) = q(S) − 1 (cf. [2], Ex. IX.1). Since q(S) = 0 by hypothesis, we must have g = q = 0. We shall now show that none of the three cases under consideration is possible.

a) First note that since p_{a}(C) > 0, C̄ := σ(C) is not a point, and so we have p_{a}(C̄) ≥ p_{a}(C). Thus, C̄ cannot be a section of π̄, for otherwise p_{a}(C̄) = 0, nor a component of a fibre, for otherwise p_{a}(C̄) ≤ 1 (cf. BPV[1], p. 150). It thus follows from Proposition 4.11b) that d(S, C) ≤ 0, which contradicts the inequality d(S̄, C̄) ≥ d(S̄, C̄) ≥ 1 which we obtain from Proposition 4.11a) and the hypothesis.

b) If Ē_{i} := σ(E_{i}) is not a component of a fibre of π̄, then the proof of a) shows that we have a contradiction. Thus, Ē_{i} must be a component of a fibre and hence must be equal to a fibre by Kodaira’s classification of elliptic fibres ([1], loc. cit.). It thus follows that Ē_{1} and Ē_{2} are disjoint, hence so are E_{1} and E_{2}, which contradicts the hypothesis.

c) By part b) we know that each Ē_{i} = σ(E_{i}) is a fibre of π̄, so the Ē_{i} are disjoint smooth elliptic curves on S̄ with (Ē_{i}.Ē_{i}) = 0 for 1 ≤ i ≤ t.

Write σ = σ_{n} ∘ ⋯ ∘ σ_{1}, where σ_{i} : S_{k−1} → S_{k} is a blowup of one point P_{k} ∈ S_{k}; thus S_{0} = S and S_{n} = S̄. Put E_{i,k} = σ_{k} ∘ ⋯ ∘ σ_{1}(E_{i}); note that all these curves are again disjoint smooth elliptic curves. It thus follows that the point P_{k} can lie on at most one elliptic curve E_{i,k}; if this is the case, then we get

E_{i,k}^{2} = (σ_{k}^{∗}(E_{i,k}).E_{i,k−1}) = ((E_{i,k−1} + C_{k}).E_{i,k−1}) = E_{i,k−1}^{2} + 1,

where C_{k} = σ_{k}^{−1}(P_{k}) denotes the exceptional curve. Thus, if we put E(k) = E_{1,k} +…+E_{t,k}, then we have shown that E(k −1)^{2} ≥ E(k)^{2} −1, for 1 ≤ k ≤ n, and so

m := ∑_{i=1}^{t} E_{i}^{2} = E(0)^{2} ≥ E(n)^{2} − n = −n.

On the other hand, since the self-intersection numbers of the canonical divisors on S and S̄ are related by the formula K2 ; S̄ = K_{S}^{2} + n, we obtain from the hypothesis (46) the inequality K2 ; S̄ > m + n ≥ 0 , which shows that S̄ is of general type.

4.3 Criteria for special type

In this section we want to establish some criteria which are useful for determining that a given surface is of special type, i.e. not of general type. This is based on the following property of surfaces of general type which is also interesting in itself.

Proposition 4.13 Let S be a surface of general type with minimal model f : S → S̄, and let D be a reduced, connected divisor with D^{2} > 0. Then we have

K2 ; S̄ ≤ \frac{(KS̄.f_{∗}D)^{2}}{(f_{∗}D)^{2}} ≤ \frac{(K_{S}.D)^{2}}{D^{2}} , (47)

and equality holds throughout if and only if K2 ; S̄ · D ≡ (K.D)^{2} · f^{∗}KS̄ and D does not contain any blow-down curve of f.

Remark 4.14 a) As the proof below shows, one can weaken the hypothesis on D slightly by allowing D to be sum of divisors of the above type. However, one cannot drop the hypothesis entirely, for otherwise one could apply it to D ∈ |K_{S}| (assuming p_{g} > 0 and K_{S}^{2} > 0), which, when S is not minimal, leads to the contradiction K2 ; S̄ ≤ K_{S}^{2} < K2 ; S̄.

b) If Z̃_{α} is a diagonal quotient surface of general type, then the above proposition yields the following upper bound on the first Chern class of the minimal model Z̄_{α} of Z̃_{α}:

(48) K2 ; Z̄_{α} ≤ \frac{8(g − 1)^{2}}{m} and hence K2 ; Z̄_{α} − K2 ; Z̃_{α} ≤ 𝕃_{α} + 12𝕊_{α}.

Indeed, if we take D = C̃_{x̄,1} + C̃_{x̄,2}, where x̄ ∉ R, then D^{2} = 2m and (K_{S}.D) = 4(g − 1), and so the first bound follows. The second follows from the first by using the formula for K_{S} of Theorem 3.7 together with equation (12).

The proof of the above proposition depends on the following useful fact.

Lemma 4.15 Let f : S → S′ be a birational morphism of smooth surfaces. Then for any divisor D ∈ Div(S) we have

(f_{∗}D)^{2} ≥ D^{2}, (49)

and equality holds if and only if D does not meet any blow-down curve C of f, i.e. (D.C) = 0 for every C such that f(C) is a point. Moreover, if D is reduced and connected and if f_{∗}D ≠ 0 (e.g. if D^{2} > 0), then we also have

(f_{∗}D.K_{S′}) ≤ (D.K_{S}). (50)

Moreover, if D does not contain any (−1)-curve on S, then equality holds in (50) if and only if D neither meets nor contains any blow-down curve of f.

Proof. Write f as f = f_{n} ∘ … ∘ f_{0}, where f_{k} : S_{k−1} → S_{k} is a blow-down map with respect to a (−1)-curve E_{k−1} on S_{k−1}. We now induct on n. If n = 1, then for any divisor D we have f^{∗}f_{∗}D = D + (D.E_{0})E_{0}, from which we obtain by the projection formula that (f_{∗}D)^{2} = (D.f^{∗}f_{∗}D) = D^{2} + (D.E_{0})^{2}. From this the inequality (49) is immediate, as is the assertion about when equality holds. Now suppose that D ≥ 0 is reduced and connected and that f_{∗}D ≠ 0. Since (f_{∗}D.K_{S′}) = (D.K_{S})−(D.E_{0}), the inequality (50) will follow once we have shown that (D.E_{0}) ≥ 0. The latter is clear if E_{0} is not a component of D, so assume the contrary, i.e. D = D_{1} + E_{0}. Note that E_{0} ≰ D_{1} since D is reduced. Moreover, D_{1} ≠ 0 since f_{∗}D ≠ 0. Thus, since D is connected we have (D_{1}.E_{0}) ≥ 1, and so (D.E_{0}) = (D_{1}.E_{0}) + E_{0}^{2} ≥ 0, as desired. Now suppose n > 1, and write f′ = f_{n} ∘ … ∘ f_{2}. Then by induction (f_{∗}D)^{2} = (f′_{∗}f_{0}_{∗}D)^{2} ≤ (f_{0}_{∗}D)^{2} ≤ D^{2}, and so the first assertion follows readily. To prove the second, we note that if D is reduced and connected and if f_{∗}D ≠ 0, then D̄ := f_{0}_{∗}D is also reduced and connected, and f′_{∗}D̄ ≠ 0, so by induction we have (f_{∗}D.K_{S′}) = (f′_{∗}D̄.K_{S′}) ≤ (D̄.K_{S_{1}}) ≤ (D.K_{S}), which proves the inequality (50).

Finally, to prove the last statement, assume that some blow-down curve C meets D. If C is not a component of D, then the assertion is clear by the above, so we may assume that every blow-down curve C which meets D is already a component. This, however, is impossible, for by hypothesis C cannot be a (−1)- curve, so there exists a blow-down map f′′ : S → S′′ (over which f factors) such that on S′′ there is a (−1) curve C′′ which meets the (−2)-curve f(C). By our assumption, C′′ must be a component of f(D). Replacing C by the proper transform of C′′, we see (by induction) that eventually there has to be a (−1)- curve contained in D, contrary to our hypothesis.

Proof of Proposition 4.13. The second inequality follows directly from Lemma 4.15, as does the fact that we have equality here if and only if D does not meet any blow-down curve.

Since K2 ; S̄ > 0, the Hodge Index Theorem (cf. [1], IV.2.15), applied to D̄ :=

K2 ; S̄ · f_{∗}D − (KS̄.f_{∗}D)^{2} · KS̄, yields D̄^{2} = (K2 ; S̄)^{2}(f_{∗}D)^{2} − (KS̄.f_{∗}D)^{2}KS̄ ≤ 0, which

proves the first inequality of (47). Furthermore, equality holds here if and only if D̄ ≡ 0. Thus, if equality holds in both places, then D cannot meet (nor contain) any blow-down curve, and then f^{∗}f_{∗}D = D, and hence the second assertion follows since the converse is trivial.

In order to be able to derive a useful criterion from the above result, we need to find reduced connected divisors D on S for which (K_{S}.D) < D^{2}. Such divisors are often obtained by considering (−2)-joins of two curves which are defined as follows.

Definition 4.16 Let S be a smooth surface. As usual, a (−2)-chain on S is a reduced connected divisor E = E_{1} + … + E_{s} consisting of (−2)-curves E_{i} such that (E_{i}.E_{i+1}) = 1 for 1 ≤ i ≤ s − 1 and (E_{i}.E_{j}) = 0 if |i − j| > 1.

If C_{1} and C_{2} are two distinct irreducible curves on S, then a (−2)-join of C_{1} to C_{2} of breadth t ≥ 0 is a reduced, connected divisor

D = C_{1} + C_{2} + E_{1} + … + E_{t},

where the E_{i} = E_{i1} + … + E_{is_{i}} are (−2)-chains joining C_{1} to C_{2}; i.e. (C_{1}.E_{i}) = (C_{1}.E_{i1}) > 0 and (C_{2}.E_{i}) = (C_{2}.E_{is_{i}}) > 0. If all the chains E_{i} meet each C_{j} transversely and of all the E_{i} are mutually disjoint, then we say that the (−2)- join is simple.

The above proposition implies that on a surface of general type, the breadth t of a (−2)-join has a bound in terms of invariants of C_{1} and C_{2} alone, and so the existence of a (−2)-join of sufficiently large breadth on S shows that the surface cannot be of general type. More precisely, we have

Proposition 4.17 Let S be a surface of general type with minimal model f : S → S̄, and let D be a (−2)-join of breadth t of two curves C_{1} and C_{2} on S. If k(C_{1}, C_{2}) := (K_{S}.(C_{1} + C_{2})) and i(C_{1}, C_{2}) := (C_{1} + C_{2})^{2}, then we have

K2 ; S̄ · (2t + i(C_{1}, C_{2})) ≤ k(C_{1}, C_{2})^{2}, (51)

and equality holds if and only if K2 ; S̄D ≡ k(D)f^{∗}KS̄ and if D is simple and does not contain any blow-down curve of f.

Proof. We first note that by the definition of D we have (K_{S}.D) = (K_{S}.(C_{1} + C_{2})) = k(C_{1}, C_{2}) and D^{2} ≥ 2t + (C_{1} + C_{2})^{2}, and that equality holds if and only if D is simple. Thus, since the inequality is trivial if 2t+i(C_{1}, C_{2}) ≤ 0, the assertion follows immediately from Proposition 4.13.

As an application of the above, we analyze the following situation which occurs in the modular curve case X = X(N); cf. [17].

Corollary 4.18 Suppose S is a surface of general type and D = C_{1}+C_{2}+E_{1}+E_{2} is a (−2)-join of breadth 2 of two (−3)-curves C_{1} and C_{2} on S with (C_{1}.C_{2}) ≥ 2. Then:

a) D neither meets nor contains any blow-down curve, and the same is true for any (−2)-chain which meets D.

b) Let E′ = C_{0} + Ẽ′ be a (−3, −2, …, −2)-chain, i.e. C_{0} is a (−3)-curve and Ẽ′ = Ẽ_{1} + … + Ẽ_{t} is a (−2)-chain such that (C_{0}.Ẽ′) = (C_{0}.Ẽ_{1}) = 1. Suppose that E′ joins C_{1} and C_{2}. If E′ meets a blow-down curve, then every (−2)-chain meeting D′ = D + E′ is already contained in D′.

Proof. a) If, as before, f : S → S̄ denotes the minimal model, then by (49) and the hypotheses we have (f_{∗}D)^{2} ≥ D^{2} ≥ 2. Now if some blow-down curve meets (or is contained in) D, then by (50) we have (KS̄.f_{∗}D) < (K_{S}.D) = 2, so (KS̄.f_{∗}D) ≤ 1. But then 2 ≤ K2 ; S̄(f_{∗}D)^{2} ≤ (KS̄.f_{∗}D)^{2} ≤ 1^{2}, contradiction.

b) Suppose the blow-down curve C meets E′; without loss of generality we may assume C^{2} = −1. Then by part a), C can only meet C_{0}. Moreover, (C.C_{0}) = 1, for otherwise f(C_{0}) is a singular curve on S̄ with (KS̄.f(C_{0})) ≤ −1, which is impossible. But then f_{∗}(C_{0}) is a (−2)-curve on S̄, and so f_{∗}D′ is a (−2)-join of f(C_{1}) and f(C_{2}) of breadth 3, and hence (f_{∗}D′)^{2} ≥ 4 = (KS̄.f_{∗}D′)^{2}. From Proposition 4.17 we thus see that K2 ; S̄ = 1 and that f_{∗}D′ ≡ 2KS̄.

Now suppose that E′′ is a (−2)-chain which meets D′ and which is not contained in D′. Then for some component E′′_{i} of E′′ we have (D′.E′′_{i}) > 0. Then Ē := f(E′′_{i}) is by part a) a (−2)-curve meeting f_{∗}D′, which gives the contradiction 0 < (f_{∗}D′.Ē) = (2KS̄.Ē) = 0.

Applying the above proposition to the chains of exceptional curves lying on a diagonal quotient surface Z̃_{α} yields the following result.

Proposition 4.19 Let x̄, ȳ ∈ R, and let D = D(x̄, ȳ) be the (−2)-join of C̃_{x̄,1} and C̃_{ȳ,2} formed by the (−2)-chains contained in the fibre ψ̃^{−1}(x̄, ȳ). Then D is simple and has breadth

t_{x̄,ȳ} := ∑_{d|eȳ,d≠eȳ} s_{eȳ−d,α}(x̄, ȳ). (52)

Moreover, i(C̃_{x̄,1}, C̃_{ȳ,2}) = C̃_{x̄,1}^{2} + C̃_{ȳ,2}^{2} + 2r_{1,α}(x̄, ȳ) and k(C_{x̄,1}, C_{ȳ,2}) = 2g_{x̄,α^{−1}} + 2g_{ȳ,α} − C̃_{x̄,1}^{2} − C̃_{ȳ,2}^{2} − 4, where C̃_{x̄,1}^{2} and C̃_{ȳ,2}^{2} are given by 3.15. Thus, if

2t_{x̄,ȳ} > k(C̃_{x̄,1}, C̃_{ȳ,2})^{2} − i(C̃_{x̄,1}, C̃_{ȳ,2}), (53)

then κ(Z̃_{α}) ≤ 1. If, in addition, p_{g} = 1 and if C̃_{x̄,1} and C̃_{ȳ,2} are (−2)-curves, then Z̃_{α} is a (blown-up) K3-surface.

Proof. Since all chains contained in the fibre are disjoint, D is simple by Lemma 3.12. Now a singularity s ∈ S_{α} of type (n, q) gives rise to a (−2)-chain if and only if \frac{n}{q} = [[2, 2, …, 2]] ⇔ q = n − 1 ⇔ \frac{ν}{d} = \frac{eȳ}{d} − 1 ⇔ s ∈ S_{eȳ−d,α}(x̄, ȳ), which proves (52). The formula for i(C̃_{x̄,1}, C̃_{ȳ,2}) follows from Corollary 3.15, and that for k(C̃_{x̄,1}, C̃_{ȳ,2}) follows from the adjunction formula, using the fact that C̃_{x̄,1} and C̃_{ȳ,2} are smooth curves of genera g_{x̄,α^{−1}} and g_{ȳ,α}, respectively (cf. Proposition 2.1). Thus, since K2 ; S̄ ≥ 1, the hypothesis (53) implies that the inequality (51) cannot hold, and hence Z̃_{α} cannot be of general type by Proposition 4.17.

Now suppose that p_{g} = 1 and that C̃_{x̄,1} and C̃_{ȳ,2} are (−2)-curves. Then κ(Z̃_{α}) ≥ 0, and Z̃_{α} is regular (because q = ḡ ≤ g_{x̄,α^{−1}} = 0). Since we already know that κ(Z̃_{α}) ≤ 1, it follows from the classification theory that the minimal model Z̄ of Z̃_{α} is either a K3-surface or an elliptic surface. Since we are done in the former case, we assume the latter, so we have a fibration f : Z̄ → ℙ^{1} (cf. proof of Theorem 4.12). Then by Kodaira’s formula we have KZ̄ ∼ ∑_{i=1}^{s}(m_{i} −1)F_{i}, where m_{1}F_{1}, …, m_{s}F_{s} are the exceptional fibres of f, because p_{g} + q − 1 = 0 (cf. proof of 4.11b)). Since no blow-down curve of Z̃_{α} → Z̄ meets D (otherwise Z̃_{α} would be rational), we may assume that D lies on Z̄. Now since D^{2} > 0, at least one component, say C, of D is not contained in any fibre of f because the intersection pairing is negative definite on the space generated by the components of the fibres.

Thus (C.F_{i}) > 0, for 1 ≤ i ≤ s, and hence 0 = (KZ̄.C) = ∑_{i=1}^{s}(m_{i} − 1)(F_{i}.C). But this means that all m_{i} = 1, so KZ̄ ∼ 0, and hence Z̄ is a K3-surface in this case as well.

Proof of Theorem 6 of the introduction: We first recall that C̃_{x̄,1} and C̃_{x̄,2} are smooth curves on Z̃_{α} of genera g_{x̄,α^{−1}} and g_{x̄,α}, respectively, with intersection numbers

C̃_{x̄,1}^{2} = −c_{x,α}, C̃_{x̄,2}^{2} = −c_{x,α^{−1}} and (C̃_{x̄,1}.C̃_{x̄,2}) = r_{1,x,α} = r_{1,α}(x̄, x̄);

cf. Proposition 2.1 and Corollary 3.15 (together with the symmetry formula (14)).

a) Here, the hypotheses mean that C̃_{x̄,1} and C̃_{x̄,2} are two intersecting (rational) (−1)-curves, and so the surface is rational (cf. van der Geer [6], VII.2.2).

b) In view of the above intersection numbers, this is just a restatement of Proposition 4.19.

c) This follows from Theorem 4.12 by taking C = C̃_{x̄,2} and E_{1} = C̃_{x̄,1}, E_{2} = C̃_{x̄,2} in 4.12a) and 4.12b), respectively.

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Ernst Kani Department of Mathematics and Statistics Queen’s University Kingston, Ontario K7L 3N6 Canada