On Asymptotic Bounds for the Number of

Irreducible Components of the Moduli Space of

Surfaces of General Type II

Michael Lönne and Matteo Penegini

Received: July 29, 2015 Revised: December 12, 2015

Communicated by Thomas Peternell

Abstract. In this paper we investigate the asymptotic growth of the number of irreducible and connected components of the moduli space of surfaces of general type corresponding to certain families of surfaces isogenous to a higher product with group (ℤ/2ℤ)^{k}. We obtain a significantly higher growth than the one in our previous paper [LP14].

2010 Mathematics Subject Classification: 14J10, 14J29, 20D15, 20D25, 20H10, 30F99. Keywords and Phrases: moduli spaces, surfaces of general type, Hurwitz action, surfaces isogenous to a product

1 Introduction

It is well known (see [Gie77]) that once two positive integers x, y are fixed there exists a quasi-projective coarse moduli space M_{y,x} of canonical models of surfaces of general type with x = χ(S) = χ(O_{S}) and y = K_{S}^{2}. The number ι(x, y), resp. γ(x, y), of irreducible, resp. connected, components of M_{y,x} is bounded from above by a function of y. In fact, Catanese proved that the number ι^{0}(x, y) of components containing regular surfaces, i.e., q(S) = 0, has an exponential upper bound in K^{2}. More precisely [Cat92, p.592] gives the following inequality

ι^{0}(x, y) ≤ y^{77y^{2}}.

This result is not known to be sharp and in recent papers [M97, Ch96, GP14, LP14] inequalities are proved which tell how close one can get to this bound from below. In particular, in the last two papers the authors considered families of surfaces isogenous to a product in order to construct many irreducible components of the moduli space of surfaces of general type. The reason why one works with these surfaces, is the fact that the number of families of these surfaces can be easily computed using group theoretical and combinatorial methods. In our previous work [LP14] we constructed many such families with many different 2−groups. There, we exploited the fact that the number of 2−groups with given order grows very fast in function of the order. In this paper we obtain a significantly better lower bound for ι^{0}(x, y) using only the groups (ℤ/2ℤ)^{k} and again some properties of the moduli space of surfaces isogenous to a product. Our main result is the following theorem.

Theorem 1.1. Let h = h_{k,l} be number of connected components of the moduli space of surfaces of general type which contain regular surfaces isogenous to a product of holomorphic Euler number x_{k,l} = 2^{l−3}(k^{2}+k−4) given by a (ℤ/2ℤ)^{k} action with ramification structure of type (2^{k(k+1)}, 2^{4+2^{l−k+1}}). If k, l are positive integers with l > 2k, then

h ≥ 2^{\sqrt[2+ν]{x_{k}}} for k → ∞,

where ν is the positive real number such that l = (2 + ν)k. In particular, given any sequence α_{i} which is positive, increasing and bounded by \frac{1}{2} from above, we obtain increasing sequences x_{i} and y_{i} = 8x_{i} with

ι^{0}(x_{i}, y_{i}) ≥ y_{i}^{(y_{i}^{α_{i}})}.

Let us explain now the way in which this paper is organized. In the next section Preliminaries we recall the definition and the properties of surfaces isogenous to a higher product and the its associated group theoretical data. Moreover, we recall a result of Bauer–Catanese [BC] which allows us to count the number of connected components of the moduli space of surfaces isogenous to a product with given group and type of ramification structure. In the last section we give the proof of the Theorem 1.1.

Acknowledgement The first author was supported by the ERC 2013 Advanced Research Grant - 340258 - TADMICAMT, at the Universität Bayreuth. The second author acknowledges Riemann Fellowship Program of the Leibniz Universität Hannover. We thank the anonymous referee for reading very carefully the manuscript and for many questions and suggestions that helped us to improve the paper a lot.

Notation and conventions. We work over the field ℂ of complex numbers. By surface we mean a projective, non-singular surface S. For such a surface ω_{S} = O_{S}(K_{S}) denotes the canonical bundle, p_{g}(S) = h^{0}(S, ω_{S}) is the geometric genus, q(S) = h^{1}(S, ω_{S}) is the irregularity, χ(O_{S}) = χ(S) = 1 − q(S) + p_{g}(S) is the Euler-Poincaré characteristic and e(S) is the topological Euler number of S.

2 Preliminaries

Definition 2.1. A surface S is said to be isogenous to a higher product of curves if and only if, S is a quotient (C_{1} × C_{2})/G, where C_{1} and C_{2} are curves of genus at least two, and G is a finite group acting freely on C_{1} × C_{2}.

Using the same notation as in Definition 2.1, let S be a surface isogenous to a product, and G^{∘} := G ∩ (Aut(C_{1}) × Aut(C_{2})). Then G^{∘} acts on the two factors C_{1}, C_{2} and diagonally on the product C_{1} × C_{2}. If G^{∘} acts faithfully on both curves, we say that S = (C_{1} × C_{2})/G is a minimal realization. In [Cat00] it is also proven that any surface isogenous to a product admits a unique minimal realization.

Assumptions. In the following we always assume:

1. Any surface S isogenous to a product is given by its unique minimal realization;

2. G^{∘} = G, this case is also known as unmixed type, see [Cat00].

Under these assumption we have.

Proposition 2.2. [Cat00] Let S = (C_{1} × C_{2})/G be a surface isogenous to a higher product of curves, then S is a minimal surface of general type with the following invariants:

χ(S) = \frac{(g(C_{1}) − 1)(g(C_{2}) − 1)}{|G|}, e(S) = 4χ(S), K_{S}^{2} = 8χ(S). (1)

The irregularity of these surfaces is computed by

q(S) = g(C_{1}/G) + g(C_{2}/G).  (2)

Among the nice features of surfaces isogenous to a product, one is that their deformation class can be obtained in a purely algebraic way. Let us briefly recall this in the particular case when S is regular, i.e., q(S) = 0, C_{i}/G ≅ ℙ^{1}.

Definition 2.3. Let G be a finite group and r ∈ ℕ with r ≥ 2.

• An r−tuple T = (v_{1}, …, v_{r}) of elements of G is called a spherical system of generators of G if ⟨v_{1}, …, v_{r}⟩ = G and v_{1} · . . . · v_{r} = 1.

• We say that T has an unordered type τ := (m_{1}, …, m_{r}) if the orders of (v_{1}, …, v_{r}) are (m_{1}, …, m_{r}) up to a permutation, namely, if there is a permutation π ∈ 𝔖_{r} such that

ord(v_{1}) = m_{π(1)}, …, ord(v_{r}) = m_{π(r)}.

• Moreover, two spherical systems T_{1} = (v_{1,1}, …, v_{1,r_{1}}) and T_{2} = (v_{2,1}, …, v_{2,r_{2}}) are said to be disjoint, if:

Σ(T_{1}) ⋂ Σ(T_{2}) = {1},  (3)

where

Σ(T_{i}) := ⋃_{g∈G} ⋃_{j=0}^{∞} ⋃_{k=1}^{r_{i}} g · v_{i,k}^{j} · g^{−1}.

We shall also use the shorthand, for example (2^{4}, 3^{2}), to indicate the tuple (2, 2, 2, 2, 3, 3).

Definition 2.4. Let 2 < r_{i} ∈ ℕ for i = 1, 2 and τ_{i} = (m_{i,1}, …, m_{i,r_{i}}) be two sequences of natural numbers such that m_{k,i} ≥ 2. A (spherical-) ramification structure of type (τ_{1}, τ_{2}) and size (r_{1}, r_{2}) for a finite group G, is a pair (T_{1}, T_{2}) of disjoint spherical systems of generators of G, whose types are τ_{i}, such that:

ℤ ∋ \frac{|G|(−2 + ∑_{l=1}^{r_{i}}(1 − \frac{1}{m_{i,l}}))}{2} + 1 ≥ 2,  for i = 1, 2.  (4)

Remark 2.5. Following e.g., the discussion in [LP14, Section 2] we obtain that the datum of the deformation class of a regular surface S isogenous to a higher product of curves of unmixed type together with its minimal realization S = (C_{1} × C_{2})/G is determined by the datum of a finite group G together with two disjoint spherical systems of generators T_{1} and T_{2} (for more details see also [BCG06]).

Remark 2.6. Recall that from Riemann Existence Theorem a finite group G acts as a group of automorphisms of some curve C of genus g such that C/G ≅ ℙ^{1} if and only if there exist integers m_{r} ≥ m_{r−1} ≥ ⋯ ≥ m_{1} ≥ 2 such that G has a spherical system of generators of type (m_{1}, …, m_{r}) and the following Riemann-Hurwitz relation holds:

2g − 2 = |G|(−2 + ∑_{i=1}^{r}(1 − \frac{1}{m_{i}})).  (5)

Remark 2.7. Note that a group G and a ramification structure determine the main numerical invariants of the surface S. Indeed, by (1) and (5) we obtain:

4χ(S) = |G| · (−2 + ∑_{k=1}^{r_{1}}(1 − \frac{1}{m_{1,k}})) · (−2 + ∑_{k=1}^{r_{2}}(1 − \frac{1}{m_{2,k}}))
=: 4χ(|G|, (τ_{1}, τ_{2})).  (6)

Let S be a surface isogenous to a product of unmixed type with group G and a pair of two disjoint spherical systems of generators of types (τ_{1}, τ_{2}). By (6) we have χ(S) = χ(G, (τ_{1}, τ_{2})), and hence, by (1), K_{S}^{2} = K^{2}(G, (τ_{1}, τ_{2})) = 8χ(S).

Let us fix a group G and a pair of unmixed ramification types (τ_{1}, τ_{2}), and denote by M_{(G,(τ_{1},τ_{2}))} the moduli space of isomorphism classes of surfaces isogenous to a product admitting these data, by [Cat00, Cat03] the space M_{(G,(τ_{1},τ_{2}))} consists of a finite number of connected components. Indeed, there is a group theoretical procedure to count these components. In case G is abelian it is described in [BC].

Theorem 2.8. [BC, Theorem 1.3] . Let S be a surface isogenous to a higher product of unmixed type and with q = 0. Then to S we attach its finite group G (up to isomorphism) and the equivalence classes of an unordered pair of disjoint spherical systems of generators (T_{1}, T_{2}) of G, under the equivalence relation generated by:

i. Hurwitz equivalence for T_{1};

ii. Hurwitz equivalence for T_{2};

iii. Simultaneous conjugation for T_{1} and T_{2}, i.e., for ϕ ∈ Aut(G) we let (T_{1} := (v_{1,1}, …, v_{1,r_{1}}), T_{2} := (v_{2,1}, …, v_{2,r_{2}})) be equivalent to

(ϕ(T_{1}) := (ϕ(v_{1,1}), …, ϕ(v_{1,r_{1}})), ϕ(T_{2}) := (ϕ(v_{2,1}), …, ϕ(v_{2,r_{2}}))).

Then two surfaces S, S′ are deformation equivalent if and only if the corresponding equivalence classes of pairs of spherical generating systems of G are the same.

The Hurwitz equivalence is defined precisely in e.g., [P13]. In the cases that we will treat the Hurwitz equivalence is given only by the braid group action on T_{i} defined as follows. Recall the Artin presentation of the Braid group of r_{1} strands

B_{r_{1}} := ⟨γ_{1}, …, γ_{r_{1}−1}| γ_{i}γ_{j} = γ_{j}γ_{i} for |i − j| ≥ 2, γ_{i+1}γ_{i}γ_{i+1} = γ_{i}γ_{i+1}γ_{i}⟩.

For γ_{i} ∈ B_{r_{1}} then:

γ_{i}(T_{1}) = γ_{i}(v_{1}, …, v_{r_{1}}) = (v_{1}, …, v_{i+1}, v_{i+1}^{−1}v_{i}v_{i+1}, …, v_{r_{1}}).

Moreover, notice that, since we deal here with abelian groups only, the braid group action is indeed only by permutation of the elements on the spherical system of generators. Once we fix a finite abelian group G and a pair of types (τ_{1}, τ_{2}) (of size (r_{1},r_{2})) of an unmixed ramification structure for G, counting the number of connected components of M_{(G,(τ_{1},τ_{2}))} is then equivalent to the group theoretical problem of counting the number of classes of pairs of spherical systems of generators of G of type (τ_{1}, τ_{2}) under the equivalence relation given by the action of B_{r_{1}} × B_{r_{2}} × Aut(G), given by:

(γ_{1}, γ_{2}, ϕ) · (T_{1}, T_{2}) :=(ϕ(γ_{1}(T_{1})), ϕ(γ_{2}(T_{2}))),  (7)

where γ_{1} ∈ B_{r_{1}}, γ_{2} ∈ B_{r_{1}} and ϕ ∈ Aut(G), see for more details e.g., [P13].

3 Proof of Theorem 1.1

Let us consider the group G := (ℤ/2ℤ)^{k}, with k >> 0 and an integer l. We want to give to G many classes of ramification structures of size (r_{1}, r_{2}) = (k(k + 1), 2^{l−k+1} + 4). Since the elements of G have only order two we will produce in the end ramification structure of type ((2^{r_{1}}), (2^{r_{2}})). Let T_{1} := (v_{1}, v_{2}, …, v_{k(k+1)}) with the following elements v_{i} of G

v_{1} = (1, 0, 0, …, 0)
v_{2} = (1, 0, 0, …, 0)
v_{3} = (0, 1, 0, …, 0)
v_{4} = (0, 1, 0, …, 0)
v_{5} = (0, 1, 0, …, 0)
v_{6} = (0, 1, 0, …, 0)
⋮
v_{k(k+1)} = (0, 0, …, 0, 1)

By construction the product of the elements in T_{1} is 1_{G} and < T_{1} >≅ G. Define the set M := G \ {0, v_{1}, …, v_{k(k+1)}}. Since M has #M = 2^{k} − k − 1 elements we can choose a bijection

φ : {n ∈ ℕ|n ≤ 2^{k} − k − 1} ⟶ M.

Let B the set of (2^{k} − k − 1)-tuples (n_{1}, …, n_{#M}) of positive integers of sum n_{1} + n_{2} + ⋯ + n_{#M} equal to 2^{l−k} + 2. For every element in B we get a 2^{l−k+1}+ 4-tuple T_{2} of elements of G by the map

(n_{1}, …, n_{2^{k}−k−1}) ↦ T_{2}

where

T_{2} = ( \underbrace{φ(1), …, φ(1)}_{2n_{1}}, …, \underbrace{φ(2^{k} − k − 1), …, φ(2^{k} − k − 1)}_{2n_{2^{k}−k−1}} ).

By construction again < T_{2} >≅ G (for k > 2) and the product of the elements in T_{2} is 1_{G}. Hence, T_{2} is a spherical system of generators for G of size 2^{l−k+1}+4. Since G = ℤ/2^{k} is abelian with all non-trivial elements of order two, the pair (T_{1}, T_{2}) is a ramification structure for G of the desired type, for any element in B.

Now we count how many inequivalent ramification structures of this kind we have under the action of the group defined in Theorem 2.8 and Equation (7). First, by construction, the only element in Aut(G), which stabilises T_{1} is the identity. Next, accordingly, (T_{1}, T′_{2}) and (T_{1}, T′′_{2}) are equivalent if and only if T′_{2} and T′′_{2} are in the same braid orbit. So finally, by construction, we get inequivalent pairs associated to different elements of B. Hence the number of inequivalent such ramification structures is equal to the number of (2^{k} − k − 1)-tuples in B of positive integers whose sum is 2^{l−k} + 2 = r_{2}/2. Any element in B corresponds uniquely to the sequence of 2^{k} − k − 1 integers n_{1}, n_{1} + n_{2}, …, n_{1} + ⋯ + n_{#M} = r_{2}/2. The sequence is strongly increasing by n_{i} > 0, so the elements in B correspond bijectively to the number of choices of 2^{k} − k − 2 integers ( all but the last integer in the sequence) in the range from 1 to r_{2}/2 − 1, cf [F68, Section II.5, p.38]. Hence,

#B = \binom{\frac{r_{2}}{2} − 1}{2^{k} − k − 2} = \binom{2^{l−k} + 1}{2^{k} − k − 2},

Let ν > 0 be a rational number and let us suppose that l = (ν + 2) · k, then we exploit Stirling’s approximation of the binomial coefficient - more exactly a corresponding lower bound. To bound the binomial coefficient \binom{m}{n} we use the lower bound (m − n)^{n} < m!/(m − n)! and an upper bound on n! related to Stirling’s formula. In fact, cf. [F68, eqn. (9.6), p.53],

n! = e^{d_{n}}n^{n+ \frac{1}{2}} e^{−n} for d_{n} = ln n! − (n + \frac{1}{2}) ln n + n

and we get an upper bound on replacing d_{n} by d_{1} = 1 due to the observation [F68, eqn. (9.9), p.53] that (d_{n}) is decreasing. Thus we obtain

\binom{2^{l−k} + 1}{2^{k} − k − 2} > \frac{(\frac{2^{l−k}+1}{2^{k}−k−2} − 1)^{2^{k}−k−2}e^{2^{k}−k−2}}{e\sqrt{(2^{k} − k − 2)}}

> (2^{νk})^{(2^{k}−k−2)} · \frac{e^{2^{k}−k−2}}{e\sqrt{(2^{k} − k − 2)}} > 2^{νk(2^{k})}. (8)

Now y_{k} = 8χ(S) = 2|G|(−2 + \frac{1}{2}r_{1})(−2 + \frac{1}{2}r_{2}) by (6) implies y_{k} = 2^{k} · 2^{l−k} · (k^{2} + k − 4) = 2^{(ν+2)k}(k^{2} + k − 4) according to l = (2 + ν)k. Hence we have

(y_{k})^{\frac{1}{ν+2}} · \frac{k}{(k^{2} + k − 4)^{\frac{1}{ν+2}}} = k2^{k}.

Using this, we obtain for k large enough in the second inequality

h > 2^{ν(y_{k})^{\frac{1}{ν+2}} ·_{\frac{k}{(k^{2}+k−4) \frac{1}{ν+2}}}} > 2^{(y \frac{1}{ν+2} ; k)}.  (9)

We can bound further for k large enough

2^{(y \frac{1}{ν+2} ; k)} > 2^{(y \frac{1}{2ν+2} ; k) \frac{ln y_{k}}{ln 2}} .  (10)

We use the identity x^{f(x)} = e^{f(x) ln x} = 2^{f(x) \frac{1}{ln 2} ln x} to get for all α < \frac{1}{2}

h > y_{k}^{(y_{k})^{α}}

if k is large enough, depending on α. □

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Michael Lönne Mathematik VIII Universität Bayreuth Universitätsstrasse 30 D-95447 Bayreuth, Germany michael.loenne@uni-bayreuth.de Matteo Penegini Dipartimento di Matematica Università degli Studi di Genova Via Dodecaneso 35 I-16146 Genova, Italy penegini@dima-unige.it