THE CLASSIFICATION OF SURFACES WITH pg = q = 1 ISOGENOUS TO A
PRODUCT OF CURVES
GIOVANNA CARNOVALE, FRANCESCO POLIZZI
Abstract. A smooth, projective surface S is said to be isogenous to a product if there exist
two smooth curves C, F and a finite group G acting freely on C × F so that S = (C × F )/G.
In this paper we classify all surfaces with pg = q = 1 which are isogenous to a product.
0. Introduction
The classification of smooth, complex surfaces S of general type with small birational in-
variants is quite a natural problem in the framework of algebraic geometry. For instance, one
may want to understand the case where the Euler characteristic χ(OS) is 1, that is, when the
geometric genus pg(S) is equal to the irregularity q(S). All surfaces of general type with these
invariants satisfy pg ≤ 4. In addition, if pg = q = 4 then the self-intersection K
S of the canon-
ical class of S is equal to 8 and S is the product of two genus 2 curves, whereas if pg = q = 3
then K2S = 6 or 8 and both cases are completely described ([CCML98], [HP02], [Pir02]). On
the other hand, surfaces of general type with pg = q = 0, 1, 2 are still far from being classified.
We refer the reader to the survey paper [BaCaPi06] for a recent account on this topic and a
comprehensive list of references.
A natural way of producing interesting examples of algebraic surfaces is to construct them as
quotients of known ones by the action of a finite group. For instance Godeaux constructed in
[Go31] the first example of surface of general type with vanishing geometric genus taking the
quotient of a general quintic surface of P3 by a free action of Z5. In line with this Beauville
proposed in [Be96, p. 118] the construction of a surface of general type with pg = q = 0, K
S = 8
as the quotient of a product of two curves C and F by the free action of a finite group G whose
order is related to the genera g(C) and g(F ) by the equality |G| = (g(C)−1)(g(F )−1). Gener-
alizing Beauville’s example we say that a surface S is isogenous to a product if S = (C ×F )/G,
for C and F smooth curves and G a finite group acting freely on C ×F . A systematic study of
these surfaces has been carried out in [Ca00]. They are of general type if and only if both g(C)
and g(F ) are greater than or equal to 2 and in this case S admits a unique minimal realization
where they are as small as possible. From now on, we tacitly assume that such a realization is
chosen, so that the genera of the curves and the group G are invariants of S. The action of G
can be seen to respect the product structure on C × F . This means that such actions fall in
two cases: the mixed one, where there exists some element in G exchanging the two factors (in
this situation C and F must be isomorphic) and the unmixed one, where G acts faithfully on
both C and F and diagonally on their product.
After [Be96], examples of surfaces isogenous to a product with pg = q = 0 appeared in [Par03]
and [BaCa03], and their complete classification was obtained in [BaCaGr06].
The next natural step is therefore the analysis of the case pg = q = 1. Surfaces of general type
with these invariants are the irregular ones with the lowest geometric genus and for this reason
it would be important to provide their complete description. So far, this has been obtained only
in the cases K2S = 2, 3 ([Ca81], [CaCi91], [CaCi93], [Pol05], [CaPi06]).
The goal of the present paper is to give the full list of surfaces with pg = q = 1 that are isoge-
nous to a product. Our work has to be seen as the sequel to the article [Pol07], which describe
Date: November 4, 2018.
2000 Mathematics Subject Classification. 14J29 (primary), 14L30, 14Q99, 20F05.
Key words and phrases. Surfaces of general type, isotrivial fibrations, actions of finite groups.
http://arxiv.org/abs/0704.0446v2
all unmixed cases with G abelian and some unmixed examples with G nonabelian. Apart from
the complete list of the genera and groups occurring, our paper contains the first examples of
surfaces of mixed type with q = 1. The mixed cases turn out to be much less frequent than
the unmixed ones and, as when pg = q = 0, they occur for only one value of the order of G.
However, in contrast with what happens when pg = q = 0, the mixed cases do not correspond
to the maximum value of |G| but appear for a rather small order, namely |G| = 16.
Our classification procedure involves arguments from both geometry and computational group
theory. We will give here a brief account on how the result is achieved.
If S is any surface isogenous to a product and satisfying pg = q then |G|, g(C), g(F ) are related
as in Beauville’s example and we have K2S = 8. Besides, if pg = q = 1 such surfaces are neces-
sarily minimal and of general type (Lemma 2.1).
If S = (C×F )/G is of unmixed type then the two projections πC : C×F −→ C, πF : C×F −→ F
induce two morphisms α : S −→ C/G, β : S −→ F/G, whose smooth fibres are isomorphic to F
and C, respectively. Moreover, the geometry of S is encoded in the geometry of the two cover-
ings h : C −→ C/G, f : F −→ F/G and the invariants of S impose strong restrictions on g(C),
g(F ) and |G|. Indeed we have 1 = q(S) = g(C/G) + g(F/G) so we may assume that E := C/G
is an elliptic curve and F/G ∼= P1. Then α : S −→ E is the Albanese morphism of S and the
genus galb of the general Albanese fibre equals g(F ). It is proven in [Pol07, Proposition 2.3] that
3 ≤ g(F ) ≤ 5; in particular this allows us to control |G|. The covers f and h are determined by
two suitable systems of generators for G, that we call V and W, respectively. Besides, in order
to obtain a free action of G on C×F and a quotient S with the desired invariants, V and W are
subject to strict conditions of combinatorial nature (Proposition 2.2). The geometry imposes
also strong restrictions on the possible W and the genus of C, so the existence of V and W and
the compatibility conditions can be verified through a computer search. It is worth mentioning
that the classification of finite groups of automorphisms acting on curves of genus lesser than or
equal to 5 could have also been retrieved from the existing literature ([Br90], [Ki03], [KuKi90],
[KuKu90]).
If S = (C × C)/G is of mixed type then the index two subgroup G◦ of G corresponding to
transformations that do not exchange the coordinates in C ×C acts faithfully on C. The quo-
tient E = C/G◦ is isomorphic to the Albanese variety of S and galb = g(C) (Proposition 2.5).
Moreover g(C) may only be 5, 7 or 9, hence |G| is at most 64 (Proposition 2.10). The cover
h : C −→ E is determined by a suitable system of generators V for G◦ and since the action of
G on C × C is required to be free, combinatorial restrictions involving the elements of V and
those of G \ G◦ have to be imposed (Proposition 2.6). Our classification is obtained by first
listing those groups G◦ for which V exists and then by looking at the admissible extensions G
of G◦. We find that the only possibility occurring is for g(C) = 5 so that |G| is necessarily 16
(Propositions 4.1, 4.2, 4.3).
In the last part of the paper we examine the structure of the subset of the moduli space
corresponding to surfaces isogenous to a product with pg = q = 1. It can be explicitly described
by calculating the number of orbits of the direct product of certain mapping class groups with
Aut(G) acting on the set (of pairs) of systems of generators (Proposition 5.1). In particular it
is possible to determine the number of irreducible connected components and their respective
dimensions, see the forthcoming article [Pe08].
Our computations were carried out by using the computer algebra program GAP4, whose data-
base includes all groups of order less than 2000, with the exception of 1024 (see [GAP4]). For
the reader’s convenience we included the scripts in the Appendix.
Now let us state the main result of this paper.
Main Theorem. Let S = (C × F )/G be a surface with pg = q = 1, isogenous to a product of
curves. Then S is minimal of general type and the occurrences for g(F ), g(C), G, the dimension
D of the moduli space and the number N of its connected components are precisely those in the
table below.
IdSmall
g(F ) = galb g(C) G Group(G) Type D N
3 3 (Z2)
2 G(4, 2) unmixed (∗) 5 1
3 5 (Z2)
3 G(8, 5) unmixed (∗) 4 1
3 5 Z2 × Z4 G(8, 2) unmixed (∗) 3 2
3 9 Z2 × Z8 G(16, 5) unmixed (∗) 2 1
3 5 D4 G(8, 3) unmixed 3 1
3 7 D6 G(12, 4) unmixed (∗∗) 3 1
3 9 Z2 ×D4 G(16, 11) unmixed 3 1
3 13 D2,12,5 G(24, 5) unmixed 2 1
3 13 Z2 ×A4 G(24, 13) unmixed 2 1
3 13 S4 G(24, 12) unmixed 2 1
3 17 Z2 ⋉ (Z2 × Z8) G(32, 9) unmixed 2 1
3 25 Z2 × S4 G(48, 48) unmixed 2 1
4 3 S3 G(6, 1) unmixed (∗∗) 4 1
4 5 D6 G(12, 4) unmixed 3 1
4 7 Z3 × S3 G(18, 3) unmixed 2 2
4 7 Z3 × S3 G(18, 3) unmixed 1 1
4 9 S4 G(24, 12) unmixed (∗∗) 2 1
4 13 S3 × S3 G(36, 10) unmixed 1 1
4 13 Z6 × S3 G(36, 12) unmixed 1 1
4 13 Z4 ⋉ (Z3)
2 G(36, 9) unmixed 1 2
4 21 A5 G(60, 5) unmixed (∗∗) 1 1
4 25 Z3 × S4 G(72, 42) unmixed 1 1
4 41 S5 G(120, 34) unmixed 1 1
5 3 D4 G(8, 3) unmixed (∗∗) 4 1
5 4 A4 G(12, 3) unmixed (∗∗) 2 2
5 5 Z4 ⋉ (Z2)
2 G(16, 3) unmixed 2 3
5 7 Z2 ×A4 G(24, 13) unmixed 2 2
5 7 Z2 ×A4 G(24, 13) unmixed 1 1
5 9 Z8 ⋉ (Z2)
2 G(32, 5) unmixed 1 1
5 9 Z2 ⋉D2,8,5 G(32, 7) unmixed 1 1
5 9 Z4 ⋉ (Z4 × Z2) G(32, 2) unmixed 1 1
5 9 Z4 ⋉ (Z2)
3 G(32, 6) unmixed 1 1
5 13 (Z2)
2 ×A4 G(48, 49) unmixed 1 1
5 17 Z4 ⋉ (Z2)
4 G(64, 32) unmixed 1 2
5 21 Z5 ⋉ (Z2)
4 G(80, 49) unmixed 1 2
5 5 D2,8,3 G(16, 8) mixed 2 1
5 5 D2,8,5 G(16, 6) mixed 2 3
5 5 Z4 ⋉ (Z2)
2 G(16, 3) mixed 2 1
Here IdSmallGroup(G) denotes the label of the group G in the GAP4 database of small groups.
The calculation of N is due to Penegini and Rollenske, see [Pe08], except for the cases marked
with (∗), which were already studied in [Pol07]. The cases marked with (∗∗) also appeared in
[Pol07], but the computation of N was missing.
This work is organized as follows.
In Section 1 we collect the basic facts about surfaces isogenous to a product, following the
treatment given by Catanese in [Ca00] and we fix the algebraic setup.
In Section 2 we apply the structure theorems of Catanese to the case pg = q = 1 and this leads
to Propositions 2.2 and 2.6, that provide the translation of our classification problem from ge-
ometry to algebra. All these results are used in Sections 3 and 4, which are the core of the
paper and give the complete lists of the occurring groups and genera in the unmixed and mixed
cases, respectively.
Finally, Section 5 is devoted to the description of the moduli spaces.
Notations and conventions. All varieties, morphisms, etc. in this article are defined over
C. By “surface” we mean a projective, non-singular surface S, and for such a surface KS
denotes the canonical class, pg(S) = h
0(S, KS) is the geometric genus, q(S) = h
1(S, KS) is the
irregularity and χ(OS) = 1 − q(S) + pg(S) is the Euler characteristic. Throughout the paper
we use the following notation for groups:
• Zn: cyclic group of order n.
• Dp,q,r = Zp ⋉Zq = 〈x, y | x
p = yq = 1, xyx−1 = yr〉: split metacyclic group of order pq.
The group D2,n,−1 is the dihedral group of order 2n and it will be denoted by Dn.
• Sn, An: symmetric, alternating group on n symbols.
• If x, y ∈ G, their commutator is defined as [x, y] = xyx−1y−1.
• If x ∈ G we denote by Intx the inner automorphism of G defined as Intx(g) = xgx
• IdSmallGroup(G) indicates the label of the group G in the GAP4 database of small
groups. For instance IdSmallGroup(D4) = G(8, 3) and this means that D4 is the third
in the list of groups of order 8.
Acknowledgements. The authors wish to thank M. Penegini and S. Rollenske for giving
them a preliminary version of [Pe08] and for kindly allowing them to include their results in
the Main Theorem. Moreover they are indebted with the referee for several valuable comments
and suggestions to improve this article.
1. Basic on surfaces isogenous to a product
In this section we collect for the reader’s convenience some basic results on groups acting on
curves and surfaces isogenous to a product, referring to [Ca00] for further details.
Definition 1.1. A complex surface S of general type is said to be isogenous to a product if there
exist two smooth curves C, F and a finite group G acting freely on C×F so that S = (C×F )/G.
There are two cases: the unmixed one, where G acts diagonally, and the mixed one, where
there exist elements of G exchanging the two factors (and then C, F are isomorphic).
In both cases, since the action of G on C × F is free, we have
K2S =
K2C×F
8(g(C) − 1)(g(F ) − 1)
χ(OS) =
χ(OC×F )
(g(C)− 1)(g(F ) − 1)
hence K2S = 8χ(OS).
Let C, F be curves of genus ≥ 2. Then the inclusion Aut(C × F ) ⊃ Aut(C) × Aut(F ) is an
equality if C and F are not isomorphic, whereas Aut(C × C) = Z2 ⋉ (Aut(C) × Aut(C)), the
Z2 being generated by the involution exchanging the two coordinates. If S = (C × F )/G is
a surface isogenous to a product, we will always consider its unique minimal realization. This
means that
• in the unmixed case, we have G ⊂ Aut(C) and G ⊂ Aut(F ) (i.e. G acts faithfully on
both C and F );
• in the mixed case, where C ∼= F , we haveG◦ ⊂ Aut(C), for G◦ := G∩(Aut(C)×Aut(C)).
(See [Ca00, Corollary 3.9 and Remark 3.10]).
Definition 1.2. Let G be a finite group and let g′ ≥ 0, and mr ≥ mr−1 ≥ . . . ≥ m1 ≥ 2 be
integers. A generating vector for G of type (g′ | m1, . . . ,mr) is a (2g
′ + r)-ple of elements
V = {g1, . . . , gr; h1, . . . , h2g′}
such that: the set V generates G; |gi| = mi and g1g2 · · · grΠ
i=1[hi, hi+g′ ] = 1. If such a V exists,
then G is said to be (g′ | m1, . . . ,mr)-generated.
For convenience we make abbreviations such as (4 | 23, 32) for (4 | 2, 2, 2, 3, 3) when we write
down the type of the generating vector V.
By Riemann’s existence theorem a finite group G acts as a group of automorphisms of some
compact Riemann surface X of genus g with quotient a Riemann surface Y of genus g′ if and
only if there exist integers mr ≥ mr−1 ≥ . . . ≥ m1 ≥ 2 such that G is (g
′ | m1, . . . ,mr)-
generated and g, g′, |G| and the mi are related by the Riemann-Hurwitz formula. Moreover,
if V = {g1, . . . , gr; h1, . . . , h2g′} is a generating vector for G, the subgroups 〈gi〉 and their
conjugates are precisely the nontrivial stabilizers of the G-action ([Br90, Section 2], [Bre00,
Chapter 3], [H71]). The description of surfaces isogenous to a product can be therefore reduced
to finding suitable generating vectors. Requiring that S has given invariants pg and q imposes
numerical restrictions on the order of the group G and the genus of the curves C and F . Our
goal is to classify all surfaces with pg = q = 1 isogenous to a product. The aim of the next
section is to translate this classification problem from geometry to algebra.
2. The case pg = q = 1. Building data
Lemma 2.1. Let S = (C × F )/G be a surface isogenous to a product with pg = q = 1. Then
(i) K2S = 8.
(ii) |G| = (g(C)− 1)(g(F ) − 1).
(iii) S is a minimal surface of general type.
Proof. Claims (i) and (ii) follow from (1). Now let us consider (iii). Since C×F is minimal and
the cover C×F −→ S is étale, S is minimal as well. Moreover (ii) implies either g(C) = g(F ) = 0
or g(C) ≥ 2, g(F ) ≥ 2. The first case is impossible otherwise S = P1 × P1 and pg = q = 0; thus
the second case occurs, hence S is of general type. �
2.1. Unmixed case. If S = (C ×F )/G is a surface with pg = q = 1, isogenous to an unmixed
product, then g(C) ≥ 3, g(F ) ≥ 3 and up to exchanging F and C one may assume F/G ∼= P1
and C/G ∼= E, where E is an elliptic curve. Moreover α : S −→ C/G is the Albanese morphism
of S and galb = g(F ), see [Pol07, Proposition 2.2]. This leads to
Proposition 2.2. ([Pol07, Proposition 3.1]) Let G be a finite group which is both (0 | m1, . . . ,mr)
and (1 | n1, . . . , ns)-generated, with generating vectors V = {g1, . . . , gr} and W = {ℓ1, . . . , ℓs; h1, h2},
respectively. Let g(F ), g(C) be the positive integers defined by the Riemann-Hurwitz relations
2g(F ) − 2 = |G|
− 2 +
2g(C)− 2 = |G|
Assume moreover that g(C) ≥ 3, g(F ) ≥ 3, |G| = (g(C) − 1)(g(F ) − 1) and
〈σgiσ
〈σℓjσ
 = {1G}.
Then there is a free, diagonal action of G on C × F such that the quotient S = (C × F )/G is
a minimal surface of general type with pg = q = 1, K
S = 8. Conversely, every surface with
pg = q = 1, isogenous to an unmixed product, arises in this way.
Here, condition (U) ensures that the G-action on C × F is free.
Set m := (m1, . . . ,mr) and n := (n1, . . . , ns); if S = (C × F )/G is a surface with pg = q = 1
which is constructed by using the recipe in Proposition 2.2, it will be called an unmixed surface
of type (G, m, n).
Proposition 2.3. ([Pol07, Proposition 2.3]) Let S = (C ×F )/G be an unmixed surface of type
(G, m, n). Then there are exactly the following possibilities:
(a) g(F ) = 3, n = (22)
(b) g(F ) = 4, n = (3)
(c) g(F ) = 5, n = (2).
The following lemma gives a restriction on m instead.
Lemma 2.4. Let S = (C × F )/G be an unmixed surface of type (G, m, n). Then every mi
divides
(g(F )−1)
Proof. Since 〈gi〉 is a stabilizer for the G-action on F and since G acts freely on (C × F ),
the subgroup 〈gi〉 ∼= Zmi acts freely on C. By Riemann-Hurwitz formula applied to the cover
C −→ C/〈gi〉 we have g(C)− 1 = mi(g(C/〈gi〉)− 1). Thus mi divides g(C)− 1 =
(g(F )−1)
2.2. Mixed case.
Proposition 2.5. Let S = (C × C)/G be a surface with pg = q = 1 isogenous to a mixed
product. Then E := C/G◦ is an elliptic curve isomorphic to the Albanese variety of S.
Proof. We have (see [Ca00, Proposition 3.15])
C = H0(Ω1S) = (H
0(Ω1C)⊕H
0(Ω1C))
G = (H0(Ω1C)
G◦ ⊕H0(Ω1C)
G◦)G/G
= (H0(Ω1E)⊕H
0(Ω1E))
G/G◦ .
Since S is of mixed type, the quotient Z2 = G/G
◦ exchanges the last two summands, whence
h0(Ω1E) = 1. Thus E is an elliptic curve and there is a commutative diagram
(3) C × C
E × E
K E(2)
showing that the Albanese morphism α of S factors through the Abel-Jacobi map α̂ of the
double symmetric product E(2) of E. �
By Lemma 2.1 we have |G| = (g(C)− 1)2. In this case [Ca00, Proposition 3.16] becomes
Proposition 2.6. Assume that G◦ is a (1 | n1, . . . , ns)-generated finite group with generating
vector V = {ℓ1, . . . , ℓs; h1, h2} and that there is a nonsplit extension
(4) 1 −→ G◦ −→ G −→ Z2 −→ 1
which gives an involution [ϕ] in Out(G◦). Let g(C) ∈ N be defined by the Riemann-Hurwitz
relation 2g(C)−2 = |G◦|
. Assume, in addition, that |G| = (g(C)−1)2 and that
(M1) for all g ∈ G \G◦ we have
{ℓ1, . . . , ℓs} ∩ {gℓ1g
−1, . . . , gℓsg
−1} = ∅;
(M2) for all g ∈ G \G◦ we have
g2 /∈
〈σℓjσ
Then there is a free, mixed action of G on C × C such that the quotient S = (C × C)/G is a
minimal surface of general type with pg = q = 1, K
S = 8.
Conversely, every surface S with pg = q = 1, isogenous to a mixed product, arises in this way.
Here, conditions (M1) and (M2) ensure that the G-action on C × C is free.
Remark 2.7. The surface S is not covered by elliptic curves because it is of general type
(Lemma 2.1), so the map C −→ C/G◦ = E is ramified. Therefore condition (M1) implies that
G is not abelian.
Remark 2.8. The exact sequence (4) is non split if and only if the number of elements of order
2 in G equals the number of elements of order 2 in G◦.
Proposition 2.9. Let S = (C × C)/G be a surface with pg = q = 1, isogenous to a mixed
product. Then galb = g(C).
Proof. Let us look at diagram (3). The Abel-Jacobi map α̂ gives to E(2) the structure of a
1-bundle over E ([CaCi93]); let f be the generic fibre of this bundle and F ∗ := ρ∗ε∗(f). If
Falb is the generic Albanese fibre of S we have Falb = π(F
∗). Let n = (n1, . . . , ns) be such
that G◦ is (1 |n1, . . . ns)-generated and 2g(C)− 2 = |G
. The (G◦ ×G◦)-cover
ρ is branched exactly along the union of s “horizontal” copies of E and s “vertical” copies of
E; moreover for each i there are one horizontal copy and one vertical copy whose branching
number is ni. Since ε
∗(f) is an elliptic curve that intersects all these copies of E transversally
in one point, by Riemann-Hurwitz formula applied to F ∗ −→ ε∗(f) we obtain
2g(F ∗)− 2 = |G◦|2 ·
On the other hand the G-cover π is étale, so we have
2g(Falb)− 2 =
(2g(F ∗)− 2) = |G◦|
= 2g(C)− 2,
whence galb = g(C). �
If S = (C × C)/G is a surface with pg = q = 1 which is constructed by using the recipe of
Proposition 2.6, it will be called a mixed surface of type (G, n). The analogue of Proposition
2.3 in the mixed case is
Proposition 2.10. Let S = (C × C)/G be a mixed surface of type (G, n). Then there are at
most the following possibilities:
• g(C) = 5, n = (22), |G| = 16;
• g(C) = 7, n = (3), |G| = 36;
• g(C) = 9, n = (2), |G| = 64.
Proof. By Proposition 2.6 we have 2g(C) − 2 = |G◦|
and |G◦| = 1
(g(C) − 1)2,
so g(C) must be odd and we obtain 4 = (g(C)− 1)
. Therefore 4 ≥ 1
(g(C)− 1)
and the only possibilities are g(C) = 3, 5, 7, 9.
The case g(C) = 3 is ruled out because G cannot be abelian by Remark 2.7.
If g(C) = 5 then
= 1, so n = (22) and |G| = 16.
If g(C) = 7 then
, so n = (3) and |G| = 36.
If g(C) = 9 then
, so n = (2) and |G| = 64. �
We will see in Section 2.10 that only the case g(C) = 5 actually occurs.
3. The unmixed case
The classification of surfaces of general type with pg = q = 1 isogenous to an unmixed prod-
uct is carried out in [Pol07] when the group G is abelian. Therefore in this section we assume
that G is nonabelian.
Following [BaCaGr06, Section 1.2], for an r-ple m = (m1, . . . ,mr) ∈ N
r we set
Θ(m) := −2 +
, α(m) :=
If S is an unmixed surface of type (G, m, n) then we necessarily have 2 ≤ m1 ≤ . . . ≤ mr and
Θ(m) > 0. Besides, by Proposition 2.2 we have α(m) =
g(F )−1
= g(C)− 1 ∈ N and by Lemma
2.4 each integer mi divides α(m). Then we get
Proposition 3.1. Let S = (C × F )/G be a surface with pg = q = 1 isogenous to an unmixed
product of type (G, m, n). Then the possibilities for m and α(m), written in the format mα(m),
lie in the set T below:
(2, 3, 7)84 , (2, 3, 8)48, (2, 4, 5)40, (2, 3, 9)36 , (2, 3, 10)30 , (2, 3, 12)24 ,
(2, 4, 6)24 , (3
2, 4)24, (2, 5
2)20, (2, 3, 18)18 , (2, 4, 8)16, (3
2, 5)15,
(2, 4, 12)12 , (2, 6
2)12, (3
2, 6)12, (3, 4
2)12, (2, 5, 10)10 , (3
2, 9)9,
(2, 82)8, (4
3)8, (3, 6
2)6, (5
3)5, (2
3, 3)12, (2
3, 4)8,
(23, 6)6, (2
2, 32)6, (2
2, 42)4, (3
4)3, (2
5)4, (2
Proof. This follows combining [BaCaGr06, Proposition 1.4] with Lemma 2.4. �
By abuse of notation, we write m ∈ T instead of mα(m) ∈ T.
Now we analyze the three cases in Proposition 2.3 separately, according to the value of g(F ).
Note that if g(F ) = 3, 4, 5 then |Aut(F )| ≤ 168, 120, 192, respectively ([Bre00, p. 91]).
Proposition 3.2. If g(F ) = 3 we have precisely the following possibilities.
IdSmall
G Group(G) m
D4 G(8, 3) (2
2, 42)
D6 G(12, 4) (2
3, 6)
Z2 ×D4 G(16, 11) (2
3, 4)
D2,12,5 G(24, 5) (2, 4, 12)
Z2 ×A4 G(24, 13) (2, 6
S4 G(24, 12) (3, 4
Z2 ⋉ (Z2 × Z8) G(32, 9) (2, 4, 8)
Z2 × S4 G(48, 48) (2, 4, 6)
Proof. Since n = (22) it follows that G is (1 | 22)-generated and by the second relation in (2)
we have |G| = 2(g(C) − 1). So we must describe all unmixed surfaces of type (G,m,n) with
m ∈ T, n = (22) and |G| = 2α(m). By a computer search through the r-tuples in Proposition
3.1 we can therefore list all possibilities, proving our statement. See the GAP4 script 1 in the
Appendix to see how this procedure applies to an explicit example.
Proposition 3.3. If g(F ) = 4 we have precisely the following possibilities.
IdSmall
G Group(G) m
S3 G(6, 1) (2
D6 G(12, 4) (2
Z3 × S3 G(18, 3) (2
2, 32)
Z3 × S3 G(18, 3) (3, 6
S4 G(24, 12) (2
3, 4)
S3 × S3 G(36, 10) (2, 6
Z6 × S3 G(36, 12) (2, 6
Z4 ⋉ (Z3)
2 G(36, 9) (3, 42)
A5 G(60, 5) (2, 5
Z3 × S4 G(72, 42) (2, 3, 12)
S5 G(120, 34) (2, 4, 5)
Proof. Since n = (3) it follows that G is (1 | 3)-generated and by the second relation in (2)
we have |G| = 3(g(C) − 1). Therefore our statement can be proven searching by computer
calculation all unmixed surfaces of type (G,m,n) with m ∈ T, n = (3), |G| = 3α(m) and
α(m) ≤ 40. �
Proposition 3.4. If g(F ) = 5 we have precisely the following possibilities.
IdSmall
G Group(G) m
D4 G(8, 3) (2
A4 G(12, 3) (3
Z4 ⋉ (Z2)
2 G(16, 3) (22, 42)
Z2 ×A4 G(24, 13) (2
2, 32)
Z2 ×A4 G(24, 13) (3, 6
Z8 ⋉ (Z2)
2 G(32, 5) (2, 82)
Z2 ⋉D2,8,5 G(32, 7) (2, 8
Z4 ⋉ (Z4 × Z2) G(32, 2) (4
Z4 ⋉ (Z2)
3 G(32, 6) (43)
2 ×A4 G(48, 49) (2, 6
Z4 ⋉ (Z2)
4 G(64, 32) (2, 4, 8)
Z5 ⋉ (Z2)
4 G(80, 49) (2, 52)
Proof. Since n = (2), it follows that G is (1 | 2)-generated and by the second relation in (2)
we have |G| = 4(g(C) − 1). Therefore our statement can be proven searching by computer
calculation all unmixed surfaces of type (G,m,n) with m ∈ T, n = (2), |G| = 4α(m) and
α(m) ≤ 48. �
4. The mixed case
In this section we use Proposition 2.6 in order to classify the surfaces with pg = q = 1
isogenous to a mixed product. By Proposition 2.10 we have g(C) = 5, 7 or 9. Let us consider
the three cases separately.
4.1. The case g(C) = 5, |G| = 16.
Proposition 4.1. If g(C) = 5, |G| = 16 we have precisely the following possibilities.
IdSmall IdSmall
G◦ Group(G◦) G Group(G)
D4 G(8, 3) D2,8,3 G(16, 8)
Z2 × Z4 G(8, 2) D2,8,5 G(16, 6)
3 G(8, 5) Z4 ⋉ (Z2)
2 G(16, 3)
Proof. In this case n = (22), so our first task is to find all nonsplit sequences of type (4) for
which G◦ is a (1 | 22)-generated group of order 8. The three abelian groups of order 8 and D4
are (1 | 22)-generated whereas the quaternion group Q8 is not.
Since Z8 has only one element ℓ of order 2, condition (M1) in Proposition 2.6 cannot be satisfied
for any choice of V. By Remark 2.7 we are left to analyze the possible embeddings of Z2 × Z4,
D4 and (Z2)
3 in nonabelian groups of order 16. The groups Z2 × Z4, D4 and (Z2)
3 have 3, 5
and 7 elements of order 2, respectively. Therefore if n2 denotes the number of elements of order
2 in G, by Remark 2.8 we must consider only those groups G of order 16 with n2 ∈ {3, 5, 7}.
The nonabelian groups of order 16 with n2 = 3 are D2,8,5, Z2 × Q8 and D4,4,−1 and they all
contain a copy of Z2 × Z4. The only nonabelian group of order 16 with n2 = 5 is D2,8,3 and
it contains a subgroup isomorphic to D4. The nonabelian groups of order 16 with n2 = 7 are
Z4 ⋉ (Z2)
2 = G(16, 3) and Z2 ⋉ Q8, and only the former contains a subgroup isomorphic to
3 (cfr. [Wi05]).
Summarizing, we are left with the following cases:
D4 D2,8,3
Z2 × Z4 D2,8,5
Z2 × Z4 Z2 ×Q8
Z2 × Z4 D4,4,−1
Z4 ⋉ (Z2)
Let us analyze them separately.
• G◦ = D4, G = D2,8,3 = 〈x, y | x
2 = y8 = 1, xyx−1 = y3 〉
We consider the subgroup G◦ := 〈x, y2〉 ∼= D4. Set ℓ1 = ℓ2 = x and h1 = h2 = y
2. Condition
(M1) holds because CG(x) = 〈x, y
4〉 ⊂ G◦. Condition (M2) is satisfied because the conjugacy
class of x in G◦ is contained in the coset x〈y2〉 while for every g ∈ yG◦ we have g2 ∈ 〈y〉.
Therefore this case occurs by Proposition 2.6.
• G◦ = Z2 × Z4, G = D2,8,5 = 〈x, y | x
2 = y8 = 1, xyx−1 = y5〉
We consider the subgroup G◦ := 〈x, y2〉 ∼= Z2 × Z4. Set ℓ1 = ℓ2 = x and h1 = h2 = y
Conditions (M1) and (M2) are verified as in the previous case, so this possibility occurs.
• G◦ = Z2 × Z4, G = Z2 ×Q8 and G
◦ = Z2 × Z4, G = D4,4,−1.
All elements of order 2 in G are central so condition (M1) cannot be satisfied and these cases
do not occur.
• G◦ = (Z2)
3, G = Z4 ⋉ (Z2)
2 = 〈x, y, z | x4 = y2 = z2 = 1, xyx−1 = yz, [x, z] = [y, z] = 1〉
We consider the subgroup G◦ := 〈y, z, x2〉 ∼= (Z2)
3. Set ℓ1 = ℓ2 = y and h1 = z, h2 = x
Condition (M1) holds because G◦ is abelian and [x, y] 6= 1. Condition (M2) is satisfied because
if g ∈ xG◦ then g2 ∈ 〈z, x2〉. Therefore this case occurs. �
4.2. The case g(C) = 7, |G| = 36.
Proposition 4.2. The case g(C) = 7, |G| = 36 does not occur.
Proof. In this case n = (3), so G◦ is a group of order 18 which is (1 | 3)-generated. There are
five groups of order 18 up to isomorphism. By computer search or direct calculation we see that
the only one which is (1 | 3)-generated is Z3 × S3 = G(18, 3). Thus G would fit into a short
exact sequence
(5) 1 −→ Z3 × S3 −→ G −→ Z2 −→ 1.
A computer search shows that the only groups of order 36 containing a subgroup isomorphic to
Z3 × S3 are G(36, 10) = S3 × S3 and G(36, 12) = Z6 × S3 (see GAP4 script 2 in the Appendix).
They contain 15 and 7 elements of order 2, respectively. On the other hand Z3 × S3 contains 3
elements of order 2, so by Remark 2.8 all possible extensions of the form (5) are split and this
case cannot occur. �
4.3. The case g(C) = 9, |G| = 64.
Proposition 4.3. The case g(C) = 9, |G| = 64 does not occur.
The proof will be the consequence of the results below. First notice that, since n = (2), the
group G◦ must be (1 | 2)-generated.
Computational Fact 4.4. There exist precisely 8 groups of order 32 which are (1 | 2)-
generated, namely G(32, t) for t ∈ {2, 4, 5, 6, 7, 8, 12, 17}. The number n2 of their elements
of order 2 is given in the table below:
t 2 4 5 6 7 8 12 17
n2(G(32, t)) 7 3 7 11 11 3 3 3
Proof. Slightly modifying the first part of GAP4 script 1 in the Appendix we easily find that
the groups of order 32 which are (1 | 2)-generated are exactly those in the statement. The
number of elements of order 2 in each case are found by a quick computer search: see again the
Appendix, GAP4 script 3. �
Computational Fact 4.5. Let t ∈ {2, 4, 5, 6, 7, 8, 12, 17}. A nonsplit extension of the form
(6) 1 −→ G(32, t) −→ G(64, s) −→ Z2 −→ 1
exists if and only if the pair (t, s) is one of the following:
(2, 9), (2, 57), (2, 59), (2, 63), (2, 64), (2, 68), (2, 70), (2, 72), (2, 76), (2, 79), (2, 81), (2, 82),
(4, 11), (4, 28), (4, 122), (4, 127), (4, 172), (4, 182),
(5, 5), (5, 9), (5, 112), (5, 113), (5, 114), (5, 132), (5, 164), (5, 165), (5, 166),
(6, 33), (6, 35),
(7, 33),
(8, 37),
(12, 7), (12, 13), (12, 14), (12, 15), (12, 16), (12, 126), (12, 127), (12, 143), (12, 156),
(12, 158), (12, 160),
(17, 28), (17, 43), (17, 45), (17, 46).
Proof. Assume t = 2. Using the GAP4 script 4 in the Appendix we find that the groups of order 64
containing a subgroup isomorphic toG(32, 2) areG(64, s) for s ∈ {8, 9, 56, 57, 58, 59, 61, 62, 63,
64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82}. By Remark 2.8 and Computa-
tional Fact 4.4, in order to detect all the groups G(64, s) fitting in some nonsplit extension of
type (6) with t = 2, it is sufficient to select from the previous list the groups containing exactly
n2 = 7 elements of order 2. This can be done with the GAP4 script 5 in the Appendix, proving
the claim in the case t = 2. The proof for the other values of t may be carried out exactly in
the same way. �
Let us denote by [G, G]2 and [G
◦, G◦]2 the subsets of elements of order 2 in [G, G] and
[G◦, G◦], respectively.
Lemma 4.6. Assume g(C) = 9 and that one of the following situations occur:
• [G,G]2 ⊆ Z(G);
• there exists some element y ∈ G \G◦ commuting with all elements in [G◦, G◦]2.
Then given any generating vector V = {ℓ1; h1, h2} of type (1 | 2) for G
◦, condition (M1) in
Proposition 2.6 cannot be satisfied.
Proof. Since ℓ1 ∈ [G
◦, G◦]2 ⊆ [G, G]2, in any of the above situations CG(ℓ1) is not contained in
G◦, so (M1) cannot hold. �
Computational Fact 4.7. Let G = G(64, s) be one of the groups appearing in the list of
Computational Fact 4.5. Then [G,G]2 is not contained in Z(G) if and only if s = 5, 33, 35, 37.
Proof. See the GAP4 script 6 in the Appendix. �
Computational facts 4.5, 4.7 and Lemma 4.6 imply that we only need to analyze the following
pairs (G◦, G):
G(32, 5) G(64, 5)
G(32, 6) G(64, 33)
G(32, 7) G(64, 33)
G(32, 6) G(64, 35)
G(32, 8) G(64, 37)
Proposition 4.8. The case G◦ = G(32, 5) does not occur.
Proof. A presentation for the group G◦ is
G◦ = 〈x, y, z | x8 = y2 = z2 = 1, [y, z] = [x, z] = 1, [x, y] = z〉.
Its derived subgroup contains exactly one element of order 2, namely z. It follows that if
{ℓ1; h1, h2} is any generating vector of type (1 | 2) for G
◦, then ℓ1 = z. Since [G
◦, G◦] is
characteristic in G◦, condition (M1) cannot be satisfied for any embedding of G◦ into G. �
By using the two instructions P:=PresentationViaCosetTable(G) and TzPrintRelators(P)
and setting in the output
x := f1, y := f2, z := f3, w := f4, v := f5, u := f6
one obtains the following presentations for G(64, 33), G(64, 35) and G(64, 37).
G(64, 33) = 〈x, y, z, w, v, u | z2 = w2 = v2 = u2 = 1, x2 = w, y2 = u,
[x, zy] = z, [x, vz] = v, [x, vu] = u,
[y, z] = [y, v] = [z, v] = [w, v] = [x, u] = 1〉
G(64, 35) = 〈x, y, z, w, v, u | w2 = v2 = u2 = 1, z2 = y2 = u, x2 = w,
[y, z] = [z, w] = u, [x, yz] = z, [x, z] = uv,
[y, v] = [z, v] = [w, v] = [x, u] = 1〉
G(64, 37) = 〈x, y, z, w, v, u | v2 = u2 = 1, w2 = z2 = y2 = u, x2 = w,
[y, z] = [z, w] = u, [x, yz] = z, [x, z] = uv,
[y, v] = [z, v] = [w, v] = 1〉.
Computational Fact 4.9. Referring to presentations (7), (8) and (9), we have the following
facts.
• The group G(64, 33) contains exactly one subgroup N1 isomorphic to G(32, 6) and one
subgroup N2 isomorphic to G(32, 7), namely
N1 := 〈x, z, w, v, u〉, N2 := 〈xy, z, w, v, u〉.
• The group G(64, 35) contains exactly two subgroups N3, N4 isomorphic to G(32, 6),
namely
N3 := 〈x, z, w, v, u〉, N4 := 〈xy, z, w, v, u〉.
• The group G(64, 37) contains exactly two subgroups N5, N6 isomorphic to G(32, 8),
namely
N5 := 〈x, z, w, v, u〉, N6 := 〈xy, z, w, v, u〉.
In addition, for every i ∈ {1, . . . , 6} we have
(a) [Ni, Ni] = 〈v, u〉 ∼= Z2 × Z2.
(b) y /∈ Ni and y commutes with all elements in [Ni, Ni].
Proof. See the GAP4 script 7 in the Appendix. �
Proposition 4.10. The cases G◦ = G(32, 6), G(32, 7), G(32, 8) do not occur.
Proof. By Lemma 4.6 and Computational Fact 4.9 it follows that, given any nonsplit extension
of type (6) with G◦ as above, condition (M1) in Proposition 2.6 cannot be satisfied. �
Summing up, we finally obtain
Proof of Proposition 4.3. It follows from Propositions 4.8 and 4.10.
5. Moduli spaces
Let Ma,b be the moduli space of smooth minimal surfaces of general type with χ(OS) =
a, K2S = b; by an important result of Gieseker, Ma,b is a quasiprojective variety for all a, b ∈ N
(see [Gie77]). Obviously, our surfaces are contained in M1,8 and we want to describe their locus
there. We denote by M(G,m,n) the moduli space of unmixed surfaces of type (G,m,n) and by
M(G,n) the moduli space of mixed surfaces of type (G,n). We know that n = (22), (3) or (2) in
the unmixed case, whereas n = (22) in the mixed one. By a general result of Catanese ([Ca00]),
both M(G,m,n) and M(G,n) consist of finitely many irreducible connected components of
M1,8, all of the same dimension. More precisely, we have
dim M(G,m,n) = r + s− 3, dim M(G,n) = s.
Consider the mapping class groups in genus zero and one:
Mod0,[r] := 〈σ1, . . . , σr | σiσi+1σi = σi+1σiσi+1,
σiσj = σjσi if |i− j| ≥ 2,
σr−1σr−2 · · · σ
1 · · · σr−2σr−1 = 1〉,
Mod1,1 := 〈tα, tβ , tγ | tαtβtα = tβtαtβ, (tαtβ)
3 = 1〉,
Mod1,[2] := 〈tα, tβ , tγ , ρ | tαtβtα = tβtαtβ, tαtγtα = tγtαtγ ,
tβtγ = tγtβ, (tαtβtγ)
4 = 1,
tαρ = ρtα, tβρ = ρtβ, tγρ = ρtγ〉.
One can prove that
Mod0,[r] : = π0 Diff
+(P1 − {p1, . . . , pr}),
Mod1,1 : = π0 Diff
+(Σ1 − {p}),
Mod1,[2] : = π0 Diff
+(Σ1 − {p, q}),
where Σ1 is the torus S
1×S1 ([Schn03], [CattMu04]). This implies that we can define actions of
these groups on the set of generating vectors for G of type (0 | m1, . . . ,mr), (1 | n) and (1 | n
respectively.
If V := {g1, . . . , gr} is of type (0 | m1, . . . ,mr) then the action is given by
gi −→ gi+1
gi+1 −→ g
i+1gigi+1
gj −→ gj if j 6= i, i+ 1.
If W := {ℓ1; h1, h2} is of type (1 | n) then
ℓ1 −→ ℓ1
h1 −→ h1
h2 −→ h2h1
ℓ1 −→ ℓ1
h1 −→ h1h
h2 −→ h2.
If W := {ℓ1, ℓ2; h1, h2} is of type (1 | n
2) then
ℓ1 −→ ℓ1
ℓ2 −→ ℓ2
h1 −→ h1
h2 −→ h2h1
ℓ1 −→ ℓ1
ℓ2 −→ ℓ2
h1 −→ h1h
h2 −→ h2
ℓ1 −→ ℓ1
ℓ2 −→ h1h
1 ℓ2h1h2h
h1 −→ h
2 ℓ1h1
h2 −→ h2
ℓ1 −→ h
1 ℓ2h1h2
ℓ2 −→ h
2 ℓ1h2h1
h1 −→ h
h2 −→ h
These are called Hurwitz moves and the induced equivalence relation on generating vectors is
said Hurwitz equivalence (see [BaCa03], [BaCaGr06], [Pol07]).
Now let B(G, m, n) be the set of pairs of generating vectors (V, W) such that the assumptions
in Proposition 2.2 are satisfied; then we denote by R the equivalence relation on B(G, m, n)
generated by Hurwitz moves on V, Hurwitz moves on W and the simultaneous action of Aut(G)
on V and W. Similarly, let B(G, n) be the set of generating vectors V such that the assump-
tions of Proposition 2.6 are satisfied; then we denote by R the equivalence relation on B(G, n)
generated by the Hurwitz moves and the action of Aut(G) on V.
Proposition 5.1. The number of irreducible components in M(G, m, n) equals the number
of R-classes in B(G, m, n). Analogously, the number of irreducible components in M(G, n)
equals the number of R-classes in B(G, n).
Proof. We can repeat exactly the same argument used in [BaCaGr06, Propositions 5.2 and 5.5];
we must just replace, where it is necessary, the mapping class group of P1 with the mapping
class group of the elliptic curve E. �
Proposition 5.1 in principle allows us to compute the number of connected components of the
moduli space in each case. In practice, this task may be too hard to be achieved by hand, but it
is not out of reach if one uses the computer. Recently, M. Penegini and S. Rollenske developed a
GAP4 script that solves this problem in a rather short time. We put the result of their calculations
in the Main Theorem (see Introduction), referring the reader to the forthcoming paper [Pe08]
for further details.
6. Appendix
In this Appendix we include, for the reader’s convenience, some of the GAP4 scripts that we
have used in our computations; all the others are similar and can be easily obtained modifying
the ones below.
Let us show how the procedure in the proof of Proposition 3.2 applies to an explicit exam-
ple, namely mα(m) = (2, 4, 12)12 . First we find all the nonabelian groups of order 24 that are
(0 | 2, 4, 12)-generated. This is done using GAP4 as below; the output tells us that there is only
one such a group, namely G = G(24, 5).
gap> # -------------- SCRIPT 1 ------------------
gap> s:=NumberSmallGroups(24);; set:=[1..s];
[1..15]
gap> for t in set do
> c:=0; G:= SmallGroup(24,t);
> Ab:=IsAbelian(G);
> for g1 in G do
> for g2 in G do
> g3:=(g1*g2)^-1;
> H:= Subgroup(G, [g1,g2]);
> if Order(g1)=2 and Order(g2)=4 and Order(g3)=12 and
> Order(H)=Order(G) and
> Ab=false then
> c:=c+1; fi;
> if Order(g1)=2 and Order(g2)=4 and Order(g3)=12 and
> Order(H)=Order(G) and
> Ab=false and c=1 then
> Print(IdSmallGroup(G)," ");
> fi; od; od; od; Print("\n");
[24,5]
By using the two instructions P:=PresentationViaCosetTable(G) and TzPrintRelators(P)
we see that G has the presentation 〈x, y | x2 = y12 = 1, xyx−1 = y5〉, hence it is isomorphic to
the metacyclic group D2,12,5.
In order to speed up further computations, we define the sets G2, G4 given by the elements of
G having order 2 and 4, respectively.
gap> G:=SmallGroup(24,5);;
gap> G2:=[];; G4:=[];;
gap> for g in G do
> if Order(g)=2 then Add(G2,g); fi;
> if Order(g)=4 then Add(G4,g); fi; od;
Then we check whether G is actually (1 | 22)-generated; if not, it should be excluded.
gap> c:=0;;
gap> for l2 in G2 do
> for h1 in G do
> for h2 in G do
> l1:=(l2*h1*h2*h1^-1*h2^-1)^-1;
> K:=Subgroup(G, [l2, h1, h2]);
> if Order(l1)=2 and Order(K)=Order(G) then
> Print(IdSmallGroup(G), " is (1 | 2,2)-generated", "\n"); c:=1; fi;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
[24,5] is (1 | 2,2)-generated
We finish the proof by checking whether the surface S actually exists; the procedure is to look
for a pair (V, W) of generating vectors for G satisfying the assumptions of Proposition 2.2.
gap> c:=0;;
gap> for g1 in G2 do
> for g2 in G4 do
> g3:=(g1*g2)^-1;
> H:=Subgroup(G, [g1, g2]);
> for l2 in G2 do
> for h1 in G do
> for h2 in G do
> l1:=(l2*h1*h2*h1^-1*h2^-1)^-1;
> K:=Subgroup(G, [l2, h1, h2]);
> Boole1:=l1 in ConjugacyClass(G, g1);
> Boole2:=l1 in ConjugacyClass(G, g2^2);
> Boole3:=l1 in ConjugacyClass(G, g3^6);
> Boole4:=l2 in ConjugacyClass(G, g1);
> Boole5:=l2 in ConjugacyClass(G, g2^2);
> Boole6:=l2 in ConjugacyClass(G, g3^6);
> if Order(g3)=12 and Order(l1)=2 and
> Order(H)=Order(G) and Order(K)=Order(G) and
> Boole1=false and Boole2=false and Boole3=false and
> Boole4=false and Boole5=false and Boole6=false then
> Print("The surface exists "); c:=1; fi;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od; Print("\n");
The surface exists
The script above can be easily modified in order to obtain the list of all admissible pairs (V, W);
for instance, one of such pairs is given by
g1 = x, g2 = xy
−1, g3 = y
ℓ1 = xy
2, ℓ2 = xy
2, h1 = y, h2 = y.
Finally, here are the GAP4 scripts used in Section 4.
gap> # -------------- SCRIPT 2 ------------------
gap> s:=NumberSmallGroups(36);; set:=[1..s];
[1..14]
gap> for t in set do
> c:=0; G:=SmallGroup(36,t);
> N:=NormalSubgroups(G);
> for G0 in N do
> if IdSmallGroup(G0)=[18,3] then
> c:=c+1; fi;
> if IdSmallGroup(G0)=[18,3] and c=1 then
> Print(IdSmallGroup(G), " ");
> fi; od; od; Print("\n");
[36,10] [36,12]
gap> # -------------- SCRIPT 3 ------------------
gap> set:=[2,4,5,6,7,8,12,17];;
gap> for t in set do
> n2:=0;
> G0:=SmallGroup(32,t);
> for g in G0 do
> if Order(g)=2 then
> n2:=n2+1; fi; od;
> Print(IdSmallGroup(G0), " "); Print(n2, " ");
> od; Print("\n");
[32,2] 7 [32,4] 3 [32,5] 7 [32,6] 11 [32,7] 11
[32,8] 3 [32,12] 3 [32,17] 3
gap> # -------------- SCRIPT 4 ------------------
gap> s:=NumberSmallGroups(64);; set:=[1..s];
[1..267]
gap> for t in set do
> c:=0; G:=SmallGroup(64,t);
> N:=NormalSubgroups(G);
> for G0 in N do
> if IdSmallGroup(G0)=[32,2] then
> c:=c+1; fi;
> if IdSmallGroup(G0)=[32,2] and c=1 then
> Print(IdSmallGroup(G), " ");
> fi; od; od; Print("\n");
[64,8] [64,9] [64,56] [64,57] [64,58] [64,59] [64,61] [64,62] [64,63] [64,64]
[64,66] [64,67] [64,68] [64,69] [64,70] [64,72] [64,73] [64,74] [64,75] [64,76]
[64,77] [64,78] [64,79] [64,80] [64,81] [64,82]
gap> # -------------- SCRIPT 5 ------------------
gap> set:=[8,9,56,57,58,59,61,62,63,64,66,67,68,69,70,
>72,73,74,75,76,77,78,79,80,81,82];;
gap> for t in set do
> n2:=0; G:=SmallGroup(64,t);
> for g in G do
> if Order(g)=2 then n2:=n2+1;
> fi; od;
> if n2=7 then
> Print(IdSmallGroup(G), " ");
> fi; od; Print("\n");
[64,9] [64,57] [64,59] [64,63] [64,64] [64,68] [64,70]
[64,72] [64,76] [64,79] [64,81] [64,82]
gap> # -------------- SCRIPT 6 ------------------
gap> set:=[5,7,9,11,13,,14,15,16,28,33,35,37,43,45,46,
>57,59,63,64,68,70,72,76,79,81,82,112,113,114, 122,126,
>127,132,143,156,158,160,164,165,166,172,182];;
gap> for t in set do
> c:=0; G:=SmallGroup(64,t);
> D:=DerivedSubgroup(G);
> for d in D do
> B:=d in Center(G);
> if Order(d)=2 and B=false then
> c:=c+1; fi;
> if Order(d)=2 and B=false and c=1 then
> Print(IdSmallGroup(G), " ");
> fi; od; od; Print("\n");
[64,5] [64,33] [64,35] [64,37]
gap> # -------------- SCRIPT 7 ------------------
gap> s:=[33, 35, 37];; I:=[1, 2, 3];;
gap> r:=[ [[32,6], [32,7]], [[32,6]], [[32,8]] ];;
> for i in I do
> G:=SmallGroup(64, s[i]); Print(IdSmallGroup(G), "\n");
> for N in NormalSubgroups(G) do
> if IdSmallGroup(N) in r[i] then
> Print(N, "="); Print(IdSmallGroup(N), " ");
> Print(DerivedSubgroup(N), "\n");
> fi; od; Print("\n"); od;
[64,33]
Group( [ f1*f2, f3, f4, f5, f6 ] )=[32,7] Group( [ f5, f6 ] )
Group( [ f1, f3, f4, f5, f6 ] )=[32,6] Group( [ f5, f6 ] )
[64,35]
Group( [ f1*f2, f3, f4, f5, f6 ] )=[32,6] Group( [ f5, f6 ] )
Group( [ f1, f3, f4, f5, f6 ] )=[32,6] Group( [ f5, f6 ] )
[64,37]
Group( [ f1*f2, f3, f4, f5, f6 ] )=[32,8] Group( [ f5, f6 ] )
Group( [ f1, f3, f4, f5, f6 ] )=[32,8] Group( [ f5, f6 ] )
References
[BaCa03] I. Bauer, F. Catanese: Some new surfaces with pg = q = 0, Proceedings of the Fano Conference
(Torino, 2002).
[BaCaGr06] I. Bauer, F. Catanese, F. Grunewald: The classification of surfaces with pg = q = 0 isogenous to
a product of curves, e-print math.AG/0610267 (2006) to appear in Pure Appl. Math Q., volume in
honour of F. Bogomolov’s 60-th birthday.
[BaCaPi06] I. Bauer, F. Catanese, R. Pignatelli: Complex surfaces of general type: some recent progress, (2006),
to appear in Global methods in complex geometry, 1–58 Springer-Verlag.
[Be96] A. Beauville: Complex algebraic surfaces, Cambridge University Press 1996.
[Bre00] T. Breuer: Characters and Automorphism groups of Compact Riemann Surfaces, Cambridge University
Press 2000.
[Br90] S. A. Broughton: Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69
(1990), 233-270.
[Ca81] F. Catanese: On a class of surfaces of general type, in Algebraic Surfaces, CIME, Liguori (1981), 269-284.
[Ca00] F. Catanese: Fibred surfaces, varieties isogenous to a product and related moduli spaces, American J.
of Math. 122 (2000), 1-44.
[CaCi91] F. Catanese and C. Ciliberto: Surfaces with pg = q = 1, Sympos. Math. XXXII (1991), 49-79.
[CaCi93] F. Catanese and C. Ciliberto: Symmetric product of elliptic curves and surfaces of general type with
pg = q = 1, J. Algebraic Geom. 2 (1993), 389-411.
[CaPi06] F. Catanese, R. Pignatelli: Fibrations of low genus I, Ann. Sci. École Norm. Sup. (4)39 (2006), 1011-
1049.
[CCML98] F. Catanese, C. Ciliberto and M. M. Lopes: Of the classification of irregular surfaces of general type
with non birational bicanonical map, Trans. Amer. Math. Soc. 350 (1998), 275-308.
[CattMu04] A. Cattabriga, M. Mulazzani: (1,1)-knots via the mapping class group of the twice punctured torus,
Adv. Geom. 4 (2004), 263-277.
[GAP4] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4 ; 2006,
http : //www.gap− system.org.
[Gie77] D. Gieseker: Global moduli for surfaces of general type, Invent. Math 43 (1977), 233-282.
[Go31] L. Godeaux: Sur une surface algébrique de genre zero et bigenre deux, Atti Accad. Naz. Lincei 14
(1931), 479-481.
[H71] W. J. Harvey: On the branch loci in Teichmüller space, Trans. Amer. Mat. Soc. 153 (1971), 387-399.
[HP02] C. Hacon, R. Pardini: Surfaces with pg = q = 3, Trans. Amer. Math. Soc. 354 no. 7 (2002), 2631-2638.
[Ki03] H. Kimura: Classification of automorphism groups, up to topological equivalence, of compact Riemann
surfaces of genus 4, J. Algebra 264 (2003), 26-54.
[KuKi90] A. Kuribayashi, H. Kimura: Automorphism groups of compact Riemann surfaces of genus five, J.
Algebra 134 (1990), no. 1, 80–103.
[KuKu90] I. Kuribayashi, A. Kuribayashi: Automorphism groups of compact Riemann surfaces of genera three
and four, J. Pure Appl. Algebra 65 (1990), no. 3, 277–292.
[Par03] R. Pardini: The classification of double planes of general type with K2S = 8 and pg = 0, J. Algebra 259
(2003) no. 3, 95-118.
[Pe08] M. Penegini: Surfaces with pg = q = 2 isogenous to a product of curves: a computational approach.
With an appendix of S. Rollenske. Work in progress
[Pir02] G. P. Pirola: Surfaces with pg = q = 3, Manuscripta Math. 108 no. 2 (2002), 163-170.
[Pol05] F. Polizzi: On surfaces of general type with pg = q = 1, K
S = 3, Collect. Math. 56, no. 2 (2005),
181-234.
[Pol07] F. Polizzi: On surfaces of general type with pg = q = 1 isogenous to a product of curves, e-print
math.AG/0601063, to appear in Comm. Algebra.
[Schn03] L. Schneps: Special loci in moduli spaces of curves. Galois groups and fundamental groups, 217–275,
Math. Sci. Res. Inst. Publ. 41, Cambridge Univ. Press, Cambridge, 2003.
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Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121
Padova, Italy.
E-mail address: carnoval@math.unipd.it
Dipartimento di Matematica, Università della Calabria, Via Pietro Bucci, 87036 Arcavacata di
Rende (CS), Italy.
E-mail address: polizzi@mat.unical.it
	0. Introduction
	1. Basic on surfaces isogenous to a product
	2. The case pg=q=1. Building data
	2.1. Unmixed case
	2.2. Mixed case
	3. The unmixed case
	4. The mixed case
	4.1. The case g(C)=5,  |G|=16
	4.2. The case g(C)=7,  |G|=36
	4.3. The case g(C)=9,  |G|=64
	5. Moduli spaces
	6. Appendix
	References

  A projective surface S is said to be isogenous to a product if there exist
two smooth curves C, F and a finite group G acting freely on C \times F so that
S=(C \times F)/G. In this paper we classify all surfaces with p_g=q=1 which are
isogenous to a product.

Introduction
The classification of smooth, complex surfaces S of general type with small birational in-
variants is quite a natural problem in the framework of algebraic geometry. For instance, one
may want to understand the case where the Euler characteristic χ(OS) is 1, that is, when the
geometric genus pg(S) is equal to the irregularity q(S). All surfaces of general type with these
invariants satisfy pg ≤ 4. In addition, if pg = q = 4 then the self-intersection K
S of the canon-
ical class of S is equal to 8 and S is the product of two genus 2 curves, whereas if pg = q = 3
then K2S = 6 or 8 and both cases are completely described ([CCML98], [HP02], [Pir02]). On
the other hand, surfaces of general type with pg = q = 0, 1, 2 are still far from being classified.
We refer the reader to the survey paper [BaCaPi06] for a recent account on this topic and a
comprehensive list of references.
A natural way of producing interesting examples of algebraic surfaces is to construct them as
quotients of known ones by the action of a finite group. For instance Godeaux constructed in
[Go31] the first example of surface of general type with vanishing geometric genus taking the
quotient of a general quintic surface of P3 by a free action of Z5. In line with this Beauville
proposed in [Be96, p. 118] the construction of a surface of general type with pg = q = 0, K
S = 8
as the quotient of a product of two curves C and F by the free action of a finite group G whose
order is related to the genera g(C) and g(F ) by the equality |G| = (g(C)−1)(g(F )−1). Gener-
alizing Beauville’s example we say that a surface S is isogenous to a product if S = (C ×F )/G,
for C and F smooth curves and G a finite group acting freely on C ×F . A systematic study of
these surfaces has been carried out in [Ca00]. They are of general type if and only if both g(C)
and g(F ) are greater than or equal to 2 and in this case S admits a unique minimal realization
where they are as small as possible. From now on, we tacitly assume that such a realization is
chosen, so that the genera of the curves and the group G are invariants of S. The action of G
can be seen to respect the product structure on C × F . This means that such actions fall in
two cases: the mixed one, where there exists some element in G exchanging the two factors (in
this situation C and F must be isomorphic) and the unmixed one, where G acts faithfully on
both C and F and diagonally on their product.
After [Be96], examples of surfaces isogenous to a product with pg = q = 0 appeared in [Par03]
and [BaCa03], and their complete classification was obtained in [BaCaGr06].
The next natural step is therefore the analysis of the case pg = q = 1. Surfaces of general type
with these invariants are the irregular ones with the lowest geometric genus and for this reason
it would be important to provide their complete description. So far, this has been obtained only
in the cases K2S = 2, 3 ([Ca81], [CaCi91], [CaCi93], [Pol05], [CaPi06]).
The goal of the present paper is to give the full list of surfaces with pg = q = 1 that are isoge-
nous to a product. Our work has to be seen as the sequel to the article [Pol07], which describe
Date: November 4, 2018.
2000 Mathematics Subject Classification. 14J29 (primary), 14L30, 14Q99, 20F05.
Key words and phrases. Surfaces of general type, isotrivial fibrations, actions of finite groups.
http://arxiv.org/abs/0704.0446v2
all unmixed cases with G abelian and some unmixed examples with G nonabelian. Apart from
the complete list of the genera and groups occurring, our paper contains the first examples of
surfaces of mixed type with q = 1. The mixed cases turn out to be much less frequent than
the unmixed ones and, as when pg = q = 0, they occur for only one value of the order of G.
However, in contrast with what happens when pg = q = 0, the mixed cases do not correspond
to the maximum value of |G| but appear for a rather small order, namely |G| = 16.
Our classification procedure involves arguments from both geometry and computational group
theory. We will give here a brief account on how the result is achieved.
If S is any surface isogenous to a product and satisfying pg = q then |G|, g(C), g(F ) are related
as in Beauville’s example and we have K2S = 8. Besides, if pg = q = 1 such surfaces are neces-
sarily minimal and of general type (Lemma 2.1).
If S = (C×F )/G is of unmixed type then the two projections πC : C×F −→ C, πF : C×F −→ F
induce two morphisms α : S −→ C/G, β : S −→ F/G, whose smooth fibres are isomorphic to F
and C, respectively. Moreover, the geometry of S is encoded in the geometry of the two cover-
ings h : C −→ C/G, f : F −→ F/G and the invariants of S impose strong restrictions on g(C),
g(F ) and |G|. Indeed we have 1 = q(S) = g(C/G) + g(F/G) so we may assume that E := C/G
is an elliptic curve and F/G ∼= P1. Then α : S −→ E is the Albanese morphism of S and the
genus galb of the general Albanese fibre equals g(F ). It is proven in [Pol07, Proposition 2.3] that
3 ≤ g(F ) ≤ 5; in particular this allows us to control |G|. The covers f and h are determined by
two suitable systems of generators for G, that we call V and W, respectively. Besides, in order
to obtain a free action of G on C×F and a quotient S with the desired invariants, V and W are
subject to strict conditions of combinatorial nature (Proposition 2.2). The geometry imposes
also strong restrictions on the possible W and the genus of C, so the existence of V and W and
the compatibility conditions can be verified through a computer search. It is worth mentioning
that the classification of finite groups of automorphisms acting on curves of genus lesser than or
equal to 5 could have also been retrieved from the existing literature ([Br90], [Ki03], [KuKi90],
[KuKu90]).
If S = (C × C)/G is of mixed type then the index two subgroup G◦ of G corresponding to
transformations that do not exchange the coordinates in C ×C acts faithfully on C. The quo-
tient E = C/G◦ is isomorphic to the Albanese variety of S and galb = g(C) (Proposition 2.5).
Moreover g(C) may only be 5, 7 or 9, hence |G| is at most 64 (Proposition 2.10). The cover
h : C −→ E is determined by a suitable system of generators V for G◦ and since the action of
G on C × C is required to be free, combinatorial restrictions involving the elements of V and
those of G \ G◦ have to be imposed (Proposition 2.6). Our classification is obtained by first
listing those groups G◦ for which V exists and then by looking at the admissible extensions G
of G◦. We find that the only possibility occurring is for g(C) = 5 so that |G| is necessarily 16
(Propositions 4.1, 4.2, 4.3).
In the last part of the paper we examine the structure of the subset of the moduli space
corresponding to surfaces isogenous to a product with pg = q = 1. It can be explicitly described
by calculating the number of orbits of the direct product of certain mapping class groups with
Aut(G) acting on the set (of pairs) of systems of generators (Proposition 5.1). In particular it
is possible to determine the number of irreducible connected components and their respective
dimensions, see the forthcoming article [Pe08].
Our computations were carried out by using the computer algebra program GAP4, whose data-
base includes all groups of order less than 2000, with the exception of 1024 (see [GAP4]). For
the reader’s convenience we included the scripts in the Appendix.
Now let us state the main result of this paper.
Main Theorem. Let S = (C × F )/G be a surface with pg = q = 1, isogenous to a product of
curves. Then S is minimal of general type and the occurrences for g(F ), g(C), G, the dimension
D of the moduli space and the number N of its connected components are precisely those in the
table below.
IdSmall
g(F ) = galb g(C) G Group(G) Type D N
3 3 (Z2)
2 G(4, 2) unmixed (∗) 5 1
3 5 (Z2)
3 G(8, 5) unmixed (∗) 4 1
3 5 Z2 × Z4 G(8, 2) unmixed (∗) 3 2
3 9 Z2 × Z8 G(16, 5) unmixed (∗) 2 1
3 5 D4 G(8, 3) unmixed 3 1
3 7 D6 G(12, 4) unmixed (∗∗) 3 1
3 9 Z2 ×D4 G(16, 11) unmixed 3 1
3 13 D2,12,5 G(24, 5) unmixed 2 1
3 13 Z2 ×A4 G(24, 13) unmixed 2 1
3 13 S4 G(24, 12) unmixed 2 1
3 17 Z2 ⋉ (Z2 × Z8) G(32, 9) unmixed 2 1
3 25 Z2 × S4 G(48, 48) unmixed 2 1
4 3 S3 G(6, 1) unmixed (∗∗) 4 1
4 5 D6 G(12, 4) unmixed 3 1
4 7 Z3 × S3 G(18, 3) unmixed 2 2
4 7 Z3 × S3 G(18, 3) unmixed 1 1
4 9 S4 G(24, 12) unmixed (∗∗) 2 1
4 13 S3 × S3 G(36, 10) unmixed 1 1
4 13 Z6 × S3 G(36, 12) unmixed 1 1
4 13 Z4 ⋉ (Z3)
2 G(36, 9) unmixed 1 2
4 21 A5 G(60, 5) unmixed (∗∗) 1 1
4 25 Z3 × S4 G(72, 42) unmixed 1 1
4 41 S5 G(120, 34) unmixed 1 1
5 3 D4 G(8, 3) unmixed (∗∗) 4 1
5 4 A4 G(12, 3) unmixed (∗∗) 2 2
5 5 Z4 ⋉ (Z2)
2 G(16, 3) unmixed 2 3
5 7 Z2 ×A4 G(24, 13) unmixed 2 2
5 7 Z2 ×A4 G(24, 13) unmixed 1 1
5 9 Z8 ⋉ (Z2)
2 G(32, 5) unmixed 1 1
5 9 Z2 ⋉D2,8,5 G(32, 7) unmixed 1 1
5 9 Z4 ⋉ (Z4 × Z2) G(32, 2) unmixed 1 1
5 9 Z4 ⋉ (Z2)
3 G(32, 6) unmixed 1 1
5 13 (Z2)
2 ×A4 G(48, 49) unmixed 1 1
5 17 Z4 ⋉ (Z2)
4 G(64, 32) unmixed 1 2
5 21 Z5 ⋉ (Z2)
4 G(80, 49) unmixed 1 2
5 5 D2,8,3 G(16, 8) mixed 2 1
5 5 D2,8,5 G(16, 6) mixed 2 3
5 5 Z4 ⋉ (Z2)
2 G(16, 3) mixed 2 1
Here IdSmallGroup(G) denotes the label of the group G in the GAP4 database of small groups.
The calculation of N is due to Penegini and Rollenske, see [Pe08], except for the cases marked
with (∗), which were already studied in [Pol07]. The cases marked with (∗∗) also appeared in
[Pol07], but the computation of N was missing.
This work is organized as follows.
In Section 1 we collect the basic facts about surfaces isogenous to a product, following the
treatment given by Catanese in [Ca00] and we fix the algebraic setup.
In Section 2 we apply the structure theorems of Catanese to the case pg = q = 1 and this leads
to Propositions 2.2 and 2.6, that provide the translation of our classification problem from ge-
ometry to algebra. All these results are used in Sections 3 and 4, which are the core of the
paper and give the complete lists of the occurring groups and genera in the unmixed and mixed
cases, respectively.
Finally, Section 5 is devoted to the description of the moduli spaces.
Notations and conventions. All varieties, morphisms, etc. in this article are defined over
C. By “surface” we mean a projective, non-singular surface S, and for such a surface KS
denotes the canonical class, pg(S) = h
0(S, KS) is the geometric genus, q(S) = h
1(S, KS) is the
irregularity and χ(OS) = 1 − q(S) + pg(S) is the Euler characteristic. Throughout the paper
we use the following notation for groups:
• Zn: cyclic group of order n.
• Dp,q,r = Zp ⋉Zq = 〈x, y | x
p = yq = 1, xyx−1 = yr〉: split metacyclic group of order pq.
The group D2,n,−1 is the dihedral group of order 2n and it will be denoted by Dn.
• Sn, An: symmetric, alternating group on n symbols.
• If x, y ∈ G, their commutator is defined as [x, y] = xyx−1y−1.
• If x ∈ G we denote by Intx the inner automorphism of G defined as Intx(g) = xgx
• IdSmallGroup(G) indicates the label of the group G in the GAP4 database of small
groups. For instance IdSmallGroup(D4) = G(8, 3) and this means that D4 is the third
in the list of groups of order 8.
Acknowledgements. The authors wish to thank M. Penegini and S. Rollenske for giving
them a preliminary version of [Pe08] and for kindly allowing them to include their results in
the Main Theorem. Moreover they are indebted with the referee for several valuable comments
and suggestions to improve this article.
1. Basic on surfaces isogenous to a product
In this section we collect for the reader’s convenience some basic results on groups acting on
curves and surfaces isogenous to a product, referring to [Ca00] for further details.
Definition 1.1. A complex surface S of general type is said to be isogenous to a product if there
exist two smooth curves C, F and a finite group G acting freely on C×F so that S = (C×F )/G.
There are two cases: the unmixed one, where G acts diagonally, and the mixed one, where
there exist elements of G exchanging the two factors (and then C, F are isomorphic).
In both cases, since the action of G on C × F is free, we have
K2S =
K2C×F
8(g(C) − 1)(g(F ) − 1)
χ(OS) =
χ(OC×F )
(g(C)− 1)(g(F ) − 1)
hence K2S = 8χ(OS).
Let C, F be curves of genus ≥ 2. Then the inclusion Aut(C × F ) ⊃ Aut(C) × Aut(F ) is an
equality if C and F are not isomorphic, whereas Aut(C × C) = Z2 ⋉ (Aut(C) × Aut(C)), the
Z2 being generated by the involution exchanging the two coordinates. If S = (C × F )/G is
a surface isogenous to a product, we will always consider its unique minimal realization. This
means that
• in the unmixed case, we have G ⊂ Aut(C) and G ⊂ Aut(F ) (i.e. G acts faithfully on
both C and F );
• in the mixed case, where C ∼= F , we haveG◦ ⊂ Aut(C), for G◦ := G∩(Aut(C)×Aut(C)).
(See [Ca00, Corollary 3.9 and Remark 3.10]).
Definition 1.2. Let G be a finite group and let g′ ≥ 0, and mr ≥ mr−1 ≥ . . . ≥ m1 ≥ 2 be
integers. A generating vector for G of type (g′ | m1, . . . ,mr) is a (2g
′ + r)-ple of elements
V = {g1, . . . , gr; h1, . . . , h2g′}
such that: the set V generates G; |gi| = mi and g1g2 · · · grΠ
i=1[hi, hi+g′ ] = 1. If such a V exists,
then G is said to be (g′ | m1, . . . ,mr)-generated.
For convenience we make abbreviations such as (4 | 23, 32) for (4 | 2, 2, 2, 3, 3) when we write
down the type of the generating vector V.
By Riemann’s existence theorem a finite group G acts as a group of automorphisms of some
compact Riemann surface X of genus g with quotient a Riemann surface Y of genus g′ if and
only if there exist integers mr ≥ mr−1 ≥ . . . ≥ m1 ≥ 2 such that G is (g
′ | m1, . . . ,mr)-
generated and g, g′, |G| and the mi are related by the Riemann-Hurwitz formula. Moreover,
if V = {g1, . . . , gr; h1, . . . , h2g′} is a generating vector for G, the subgroups 〈gi〉 and their
conjugates are precisely the nontrivial stabilizers of the G-action ([Br90, Section 2], [Bre00,
Chapter 3], [H71]). The description of surfaces isogenous to a product can be therefore reduced
to finding suitable generating vectors. Requiring that S has given invariants pg and q imposes
numerical restrictions on the order of the group G and the genus of the curves C and F . Our
goal is to classify all surfaces with pg = q = 1 isogenous to a product. The aim of the next
section is to translate this classification problem from geometry to algebra.
2. The case pg = q = 1. Building data
Lemma 2.1. Let S = (C × F )/G be a surface isogenous to a product with pg = q = 1. Then
(i) K2S = 8.
(ii) |G| = (g(C)− 1)(g(F ) − 1).
(iii) S is a minimal surface of general type.
Proof. Claims (i) and (ii) follow from (1). Now let us consider (iii). Since C×F is minimal and
the cover C×F −→ S is étale, S is minimal as well. Moreover (ii) implies either g(C) = g(F ) = 0
or g(C) ≥ 2, g(F ) ≥ 2. The first case is impossible otherwise S = P1 × P1 and pg = q = 0; thus
the second case occurs, hence S is of general type. �
2.1. Unmixed case. If S = (C ×F )/G is a surface with pg = q = 1, isogenous to an unmixed
product, then g(C) ≥ 3, g(F ) ≥ 3 and up to exchanging F and C one may assume F/G ∼= P1
and C/G ∼= E, where E is an elliptic curve. Moreover α : S −→ C/G is the Albanese morphism
of S and galb = g(F ), see [Pol07, Proposition 2.2]. This leads to
Proposition 2.2. ([Pol07, Proposition 3.1]) Let G be a finite group which is both (0 | m1, . . . ,mr)
and (1 | n1, . . . , ns)-generated, with generating vectors V = {g1, . . . , gr} and W = {ℓ1, . . . , ℓs; h1, h2},
respectively. Let g(F ), g(C) be the positive integers defined by the Riemann-Hurwitz relations
2g(F ) − 2 = |G|
− 2 +
2g(C)− 2 = |G|
Assume moreover that g(C) ≥ 3, g(F ) ≥ 3, |G| = (g(C) − 1)(g(F ) − 1) and
〈σgiσ
〈σℓjσ
 = {1G}.
Then there is a free, diagonal action of G on C × F such that the quotient S = (C × F )/G is
a minimal surface of general type with pg = q = 1, K
S = 8. Conversely, every surface with
pg = q = 1, isogenous to an unmixed product, arises in this way.
Here, condition (U) ensures that the G-action on C × F is free.
Set m := (m1, . . . ,mr) and n := (n1, . . . , ns); if S = (C × F )/G is a surface with pg = q = 1
which is constructed by using the recipe in Proposition 2.2, it will be called an unmixed surface
of type (G, m, n).
Proposition 2.3. ([Pol07, Proposition 2.3]) Let S = (C ×F )/G be an unmixed surface of type
(G, m, n). Then there are exactly the following possibilities:
(a) g(F ) = 3, n = (22)
(b) g(F ) = 4, n = (3)
(c) g(F ) = 5, n = (2).
The following lemma gives a restriction on m instead.
Lemma 2.4. Let S = (C × F )/G be an unmixed surface of type (G, m, n). Then every mi
divides
(g(F )−1)
Proof. Since 〈gi〉 is a stabilizer for the G-action on F and since G acts freely on (C × F ),
the subgroup 〈gi〉 ∼= Zmi acts freely on C. By Riemann-Hurwitz formula applied to the cover
C −→ C/〈gi〉 we have g(C)− 1 = mi(g(C/〈gi〉)− 1). Thus mi divides g(C)− 1 =
(g(F )−1)
2.2. Mixed case.
Proposition 2.5. Let S = (C × C)/G be a surface with pg = q = 1 isogenous to a mixed
product. Then E := C/G◦ is an elliptic curve isomorphic to the Albanese variety of S.
Proof. We have (see [Ca00, Proposition 3.15])
C = H0(Ω1S) = (H
0(Ω1C)⊕H
0(Ω1C))
G = (H0(Ω1C)
G◦ ⊕H0(Ω1C)
G◦)G/G
= (H0(Ω1E)⊕H
0(Ω1E))
G/G◦ .
Since S is of mixed type, the quotient Z2 = G/G
◦ exchanges the last two summands, whence
h0(Ω1E) = 1. Thus E is an elliptic curve and there is a commutative diagram
(3) C × C
E × E
K E(2)
showing that the Albanese morphism α of S factors through the Abel-Jacobi map α̂ of the
double symmetric product E(2) of E. �
By Lemma 2.1 we have |G| = (g(C)− 1)2. In this case [Ca00, Proposition 3.16] becomes
Proposition 2.6. Assume that G◦ is a (1 | n1, . . . , ns)-generated finite group with generating
vector V = {ℓ1, . . . , ℓs; h1, h2} and that there is a nonsplit extension
(4) 1 −→ G◦ −→ G −→ Z2 −→ 1
which gives an involution [ϕ] in Out(G◦). Let g(C) ∈ N be defined by the Riemann-Hurwitz
relation 2g(C)−2 = |G◦|
. Assume, in addition, that |G| = (g(C)−1)2 and that
(M1) for all g ∈ G \G◦ we have
{ℓ1, . . . , ℓs} ∩ {gℓ1g
−1, . . . , gℓsg
−1} = ∅;
(M2) for all g ∈ G \G◦ we have
g2 /∈
〈σℓjσ
Then there is a free, mixed action of G on C × C such that the quotient S = (C × C)/G is a
minimal surface of general type with pg = q = 1, K
S = 8.
Conversely, every surface S with pg = q = 1, isogenous to a mixed product, arises in this way.
Here, conditions (M1) and (M2) ensure that the G-action on C × C is free.
Remark 2.7. The surface S is not covered by elliptic curves because it is of general type
(Lemma 2.1), so the map C −→ C/G◦ = E is ramified. Therefore condition (M1) implies that
G is not abelian.
Remark 2.8. The exact sequence (4) is non split if and only if the number of elements of order
2 in G equals the number of elements of order 2 in G◦.
Proposition 2.9. Let S = (C × C)/G be a surface with pg = q = 1, isogenous to a mixed
product. Then galb = g(C).
Proof. Let us look at diagram (3). The Abel-Jacobi map α̂ gives to E(2) the structure of a
1-bundle over E ([CaCi93]); let f be the generic fibre of this bundle and F ∗ := ρ∗ε∗(f). If
Falb is the generic Albanese fibre of S we have Falb = π(F
∗). Let n = (n1, . . . , ns) be such
that G◦ is (1 |n1, . . . ns)-generated and 2g(C)− 2 = |G
. The (G◦ ×G◦)-cover
ρ is branched exactly along the union of s “horizontal” copies of E and s “vertical” copies of
E; moreover for each i there are one horizontal copy and one vertical copy whose branching
number is ni. Since ε
∗(f) is an elliptic curve that intersects all these copies of E transversally
in one point, by Riemann-Hurwitz formula applied to F ∗ −→ ε∗(f) we obtain
2g(F ∗)− 2 = |G◦|2 ·
On the other hand the G-cover π is étale, so we have
2g(Falb)− 2 =
(2g(F ∗)− 2) = |G◦|
= 2g(C)− 2,
whence galb = g(C). �
If S = (C × C)/G is a surface with pg = q = 1 which is constructed by using the recipe of
Proposition 2.6, it will be called a mixed surface of type (G, n). The analogue of Proposition
2.3 in the mixed case is
Proposition 2.10. Let S = (C × C)/G be a mixed surface of type (G, n). Then there are at
most the following possibilities:
• g(C) = 5, n = (22), |G| = 16;
• g(C) = 7, n = (3), |G| = 36;
• g(C) = 9, n = (2), |G| = 64.
Proof. By Proposition 2.6 we have 2g(C) − 2 = |G◦|
and |G◦| = 1
(g(C) − 1)2,
so g(C) must be odd and we obtain 4 = (g(C)− 1)
. Therefore 4 ≥ 1
(g(C)− 1)
and the only possibilities are g(C) = 3, 5, 7, 9.
The case g(C) = 3 is ruled out because G cannot be abelian by Remark 2.7.
If g(C) = 5 then
= 1, so n = (22) and |G| = 16.
If g(C) = 7 then
, so n = (3) and |G| = 36.
If g(C) = 9 then
, so n = (2) and |G| = 64. �
We will see in Section 2.10 that only the case g(C) = 5 actually occurs.
3. The unmixed case
The classification of surfaces of general type with pg = q = 1 isogenous to an unmixed prod-
uct is carried out in [Pol07] when the group G is abelian. Therefore in this section we assume
that G is nonabelian.
Following [BaCaGr06, Section 1.2], for an r-ple m = (m1, . . . ,mr) ∈ N
r we set
Θ(m) := −2 +
, α(m) :=
If S is an unmixed surface of type (G, m, n) then we necessarily have 2 ≤ m1 ≤ . . . ≤ mr and
Θ(m) > 0. Besides, by Proposition 2.2 we have α(m) =
g(F )−1
= g(C)− 1 ∈ N and by Lemma
2.4 each integer mi divides α(m). Then we get
Proposition 3.1. Let S = (C × F )/G be a surface with pg = q = 1 isogenous to an unmixed
product of type (G, m, n). Then the possibilities for m and α(m), written in the format mα(m),
lie in the set T below:
(2, 3, 7)84 , (2, 3, 8)48, (2, 4, 5)40, (2, 3, 9)36 , (2, 3, 10)30 , (2, 3, 12)24 ,
(2, 4, 6)24 , (3
2, 4)24, (2, 5
2)20, (2, 3, 18)18 , (2, 4, 8)16, (3
2, 5)15,
(2, 4, 12)12 , (2, 6
2)12, (3
2, 6)12, (3, 4
2)12, (2, 5, 10)10 , (3
2, 9)9,
(2, 82)8, (4
3)8, (3, 6
2)6, (5
3)5, (2
3, 3)12, (2
3, 4)8,
(23, 6)6, (2
2, 32)6, (2
2, 42)4, (3
4)3, (2
5)4, (2
Proof. This follows combining [BaCaGr06, Proposition 1.4] with Lemma 2.4. �
By abuse of notation, we write m ∈ T instead of mα(m) ∈ T.
Now we analyze the three cases in Proposition 2.3 separately, according to the value of g(F ).
Note that if g(F ) = 3, 4, 5 then |Aut(F )| ≤ 168, 120, 192, respectively ([Bre00, p. 91]).
Proposition 3.2. If g(F ) = 3 we have precisely the following possibilities.
IdSmall
G Group(G) m
D4 G(8, 3) (2
2, 42)
D6 G(12, 4) (2
3, 6)
Z2 ×D4 G(16, 11) (2
3, 4)
D2,12,5 G(24, 5) (2, 4, 12)
Z2 ×A4 G(24, 13) (2, 6
S4 G(24, 12) (3, 4
Z2 ⋉ (Z2 × Z8) G(32, 9) (2, 4, 8)
Z2 × S4 G(48, 48) (2, 4, 6)
Proof. Since n = (22) it follows that G is (1 | 22)-generated and by the second relation in (2)
we have |G| = 2(g(C) − 1). So we must describe all unmixed surfaces of type (G,m,n) with
m ∈ T, n = (22) and |G| = 2α(m). By a computer search through the r-tuples in Proposition
3.1 we can therefore list all possibilities, proving our statement. See the GAP4 script 1 in the
Appendix to see how this procedure applies to an explicit example.
Proposition 3.3. If g(F ) = 4 we have precisely the following possibilities.
IdSmall
G Group(G) m
S3 G(6, 1) (2
D6 G(12, 4) (2
Z3 × S3 G(18, 3) (2
2, 32)
Z3 × S3 G(18, 3) (3, 6
S4 G(24, 12) (2
3, 4)
S3 × S3 G(36, 10) (2, 6
Z6 × S3 G(36, 12) (2, 6
Z4 ⋉ (Z3)
2 G(36, 9) (3, 42)
A5 G(60, 5) (2, 5
Z3 × S4 G(72, 42) (2, 3, 12)
S5 G(120, 34) (2, 4, 5)
Proof. Since n = (3) it follows that G is (1 | 3)-generated and by the second relation in (2)
we have |G| = 3(g(C) − 1). Therefore our statement can be proven searching by computer
calculation all unmixed surfaces of type (G,m,n) with m ∈ T, n = (3), |G| = 3α(m) and
α(m) ≤ 40. �
Proposition 3.4. If g(F ) = 5 we have precisely the following possibilities.
IdSmall
G Group(G) m
D4 G(8, 3) (2
A4 G(12, 3) (3
Z4 ⋉ (Z2)
2 G(16, 3) (22, 42)
Z2 ×A4 G(24, 13) (2
2, 32)
Z2 ×A4 G(24, 13) (3, 6
Z8 ⋉ (Z2)
2 G(32, 5) (2, 82)
Z2 ⋉D2,8,5 G(32, 7) (2, 8
Z4 ⋉ (Z4 × Z2) G(32, 2) (4
Z4 ⋉ (Z2)
3 G(32, 6) (43)
2 ×A4 G(48, 49) (2, 6
Z4 ⋉ (Z2)
4 G(64, 32) (2, 4, 8)
Z5 ⋉ (Z2)
4 G(80, 49) (2, 52)
Proof. Since n = (2), it follows that G is (1 | 2)-generated and by the second relation in (2)
we have |G| = 4(g(C) − 1). Therefore our statement can be proven searching by computer
calculation all unmixed surfaces of type (G,m,n) with m ∈ T, n = (2), |G| = 4α(m) and
α(m) ≤ 48. �
4. The mixed case
In this section we use Proposition 2.6 in order to classify the surfaces with pg = q = 1
isogenous to a mixed product. By Proposition 2.10 we have g(C) = 5, 7 or 9. Let us consider
the three cases separately.
4.1. The case g(C) = 5, |G| = 16.
Proposition 4.1. If g(C) = 5, |G| = 16 we have precisely the following possibilities.
IdSmall IdSmall
G◦ Group(G◦) G Group(G)
D4 G(8, 3) D2,8,3 G(16, 8)
Z2 × Z4 G(8, 2) D2,8,5 G(16, 6)
3 G(8, 5) Z4 ⋉ (Z2)
2 G(16, 3)
Proof. In this case n = (22), so our first task is to find all nonsplit sequences of type (4) for
which G◦ is a (1 | 22)-generated group of order 8. The three abelian groups of order 8 and D4
are (1 | 22)-generated whereas the quaternion group Q8 is not.
Since Z8 has only one element ℓ of order 2, condition (M1) in Proposition 2.6 cannot be satisfied
for any choice of V. By Remark 2.7 we are left to analyze the possible embeddings of Z2 × Z4,
D4 and (Z2)
3 in nonabelian groups of order 16. The groups Z2 × Z4, D4 and (Z2)
3 have 3, 5
and 7 elements of order 2, respectively. Therefore if n2 denotes the number of elements of order
2 in G, by Remark 2.8 we must consider only those groups G of order 16 with n2 ∈ {3, 5, 7}.
The nonabelian groups of order 16 with n2 = 3 are D2,8,5, Z2 × Q8 and D4,4,−1 and they all
contain a copy of Z2 × Z4. The only nonabelian group of order 16 with n2 = 5 is D2,8,3 and
it contains a subgroup isomorphic to D4. The nonabelian groups of order 16 with n2 = 7 are
Z4 ⋉ (Z2)
2 = G(16, 3) and Z2 ⋉ Q8, and only the former contains a subgroup isomorphic to
3 (cfr. [Wi05]).
Summarizing, we are left with the following cases:
D4 D2,8,3
Z2 × Z4 D2,8,5
Z2 × Z4 Z2 ×Q8
Z2 × Z4 D4,4,−1
Z4 ⋉ (Z2)
Let us analyze them separately.
• G◦ = D4, G = D2,8,3 = 〈x, y | x
2 = y8 = 1, xyx−1 = y3 〉
We consider the subgroup G◦ := 〈x, y2〉 ∼= D4. Set ℓ1 = ℓ2 = x and h1 = h2 = y
2. Condition
(M1) holds because CG(x) = 〈x, y
4〉 ⊂ G◦. Condition (M2) is satisfied because the conjugacy
class of x in G◦ is contained in the coset x〈y2〉 while for every g ∈ yG◦ we have g2 ∈ 〈y〉.
Therefore this case occurs by Proposition 2.6.
• G◦ = Z2 × Z4, G = D2,8,5 = 〈x, y | x
2 = y8 = 1, xyx−1 = y5〉
We consider the subgroup G◦ := 〈x, y2〉 ∼= Z2 × Z4. Set ℓ1 = ℓ2 = x and h1 = h2 = y
Conditions (M1) and (M2) are verified as in the previous case, so this possibility occurs.
• G◦ = Z2 × Z4, G = Z2 ×Q8 and G
◦ = Z2 × Z4, G = D4,4,−1.
All elements of order 2 in G are central so condition (M1) cannot be satisfied and these cases
do not occur.
• G◦ = (Z2)
3, G = Z4 ⋉ (Z2)
2 = 〈x, y, z | x4 = y2 = z2 = 1, xyx−1 = yz, [x, z] = [y, z] = 1〉
We consider the subgroup G◦ := 〈y, z, x2〉 ∼= (Z2)
3. Set ℓ1 = ℓ2 = y and h1 = z, h2 = x
Condition (M1) holds because G◦ is abelian and [x, y] 6= 1. Condition (M2) is satisfied because
if g ∈ xG◦ then g2 ∈ 〈z, x2〉. Therefore this case occurs. �
4.2. The case g(C) = 7, |G| = 36.
Proposition 4.2. The case g(C) = 7, |G| = 36 does not occur.
Proof. In this case n = (3), so G◦ is a group of order 18 which is (1 | 3)-generated. There are
five groups of order 18 up to isomorphism. By computer search or direct calculation we see that
the only one which is (1 | 3)-generated is Z3 × S3 = G(18, 3). Thus G would fit into a short
exact sequence
(5) 1 −→ Z3 × S3 −→ G −→ Z2 −→ 1.
A computer search shows that the only groups of order 36 containing a subgroup isomorphic to
Z3 × S3 are G(36, 10) = S3 × S3 and G(36, 12) = Z6 × S3 (see GAP4 script 2 in the Appendix).
They contain 15 and 7 elements of order 2, respectively. On the other hand Z3 × S3 contains 3
elements of order 2, so by Remark 2.8 all possible extensions of the form (5) are split and this
case cannot occur. �
4.3. The case g(C) = 9, |G| = 64.
Proposition 4.3. The case g(C) = 9, |G| = 64 does not occur.
The proof will be the consequence of the results below. First notice that, since n = (2), the
group G◦ must be (1 | 2)-generated.
Computational Fact 4.4. There exist precisely 8 groups of order 32 which are (1 | 2)-
generated, namely G(32, t) for t ∈ {2, 4, 5, 6, 7, 8, 12, 17}. The number n2 of their elements
of order 2 is given in the table below:
t 2 4 5 6 7 8 12 17
n2(G(32, t)) 7 3 7 11 11 3 3 3
Proof. Slightly modifying the first part of GAP4 script 1 in the Appendix we easily find that
the groups of order 32 which are (1 | 2)-generated are exactly those in the statement. The
number of elements of order 2 in each case are found by a quick computer search: see again the
Appendix, GAP4 script 3. �
Computational Fact 4.5. Let t ∈ {2, 4, 5, 6, 7, 8, 12, 17}. A nonsplit extension of the form
(6) 1 −→ G(32, t) −→ G(64, s) −→ Z2 −→ 1
exists if and only if the pair (t, s) is one of the following:
(2, 9), (2, 57), (2, 59), (2, 63), (2, 64), (2, 68), (2, 70), (2, 72), (2, 76), (2, 79), (2, 81), (2, 82),
(4, 11), (4, 28), (4, 122), (4, 127), (4, 172), (4, 182),
(5, 5), (5, 9), (5, 112), (5, 113), (5, 114), (5, 132), (5, 164), (5, 165), (5, 166),
(6, 33), (6, 35),
(7, 33),
(8, 37),
(12, 7), (12, 13), (12, 14), (12, 15), (12, 16), (12, 126), (12, 127), (12, 143), (12, 156),
(12, 158), (12, 160),
(17, 28), (17, 43), (17, 45), (17, 46).
Proof. Assume t = 2. Using the GAP4 script 4 in the Appendix we find that the groups of order 64
containing a subgroup isomorphic toG(32, 2) areG(64, s) for s ∈ {8, 9, 56, 57, 58, 59, 61, 62, 63,
64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82}. By Remark 2.8 and Computa-
tional Fact 4.4, in order to detect all the groups G(64, s) fitting in some nonsplit extension of
type (6) with t = 2, it is sufficient to select from the previous list the groups containing exactly
n2 = 7 elements of order 2. This can be done with the GAP4 script 5 in the Appendix, proving
the claim in the case t = 2. The proof for the other values of t may be carried out exactly in
the same way. �
Let us denote by [G, G]2 and [G
◦, G◦]2 the subsets of elements of order 2 in [G, G] and
[G◦, G◦], respectively.
Lemma 4.6. Assume g(C) = 9 and that one of the following situations occur:
• [G,G]2 ⊆ Z(G);
• there exists some element y ∈ G \G◦ commuting with all elements in [G◦, G◦]2.
Then given any generating vector V = {ℓ1; h1, h2} of type (1 | 2) for G
◦, condition (M1) in
Proposition 2.6 cannot be satisfied.
Proof. Since ℓ1 ∈ [G
◦, G◦]2 ⊆ [G, G]2, in any of the above situations CG(ℓ1) is not contained in
G◦, so (M1) cannot hold. �
Computational Fact 4.7. Let G = G(64, s) be one of the groups appearing in the list of
Computational Fact 4.5. Then [G,G]2 is not contained in Z(G) if and only if s = 5, 33, 35, 37.
Proof. See the GAP4 script 6 in the Appendix. �
Computational facts 4.5, 4.7 and Lemma 4.6 imply that we only need to analyze the following
pairs (G◦, G):
G(32, 5) G(64, 5)
G(32, 6) G(64, 33)
G(32, 7) G(64, 33)
G(32, 6) G(64, 35)
G(32, 8) G(64, 37)
Proposition 4.8. The case G◦ = G(32, 5) does not occur.
Proof. A presentation for the group G◦ is
G◦ = 〈x, y, z | x8 = y2 = z2 = 1, [y, z] = [x, z] = 1, [x, y] = z〉.
Its derived subgroup contains exactly one element of order 2, namely z. It follows that if
{ℓ1; h1, h2} is any generating vector of type (1 | 2) for G
◦, then ℓ1 = z. Since [G
◦, G◦] is
characteristic in G◦, condition (M1) cannot be satisfied for any embedding of G◦ into G. �
By using the two instructions P:=PresentationViaCosetTable(G) and TzPrintRelators(P)
and setting in the output
x := f1, y := f2, z := f3, w := f4, v := f5, u := f6
one obtains the following presentations for G(64, 33), G(64, 35) and G(64, 37).
G(64, 33) = 〈x, y, z, w, v, u | z2 = w2 = v2 = u2 = 1, x2 = w, y2 = u,
[x, zy] = z, [x, vz] = v, [x, vu] = u,
[y, z] = [y, v] = [z, v] = [w, v] = [x, u] = 1〉
G(64, 35) = 〈x, y, z, w, v, u | w2 = v2 = u2 = 1, z2 = y2 = u, x2 = w,
[y, z] = [z, w] = u, [x, yz] = z, [x, z] = uv,
[y, v] = [z, v] = [w, v] = [x, u] = 1〉
G(64, 37) = 〈x, y, z, w, v, u | v2 = u2 = 1, w2 = z2 = y2 = u, x2 = w,
[y, z] = [z, w] = u, [x, yz] = z, [x, z] = uv,
[y, v] = [z, v] = [w, v] = 1〉.
Computational Fact 4.9. Referring to presentations (7), (8) and (9), we have the following
facts.
• The group G(64, 33) contains exactly one subgroup N1 isomorphic to G(32, 6) and one
subgroup N2 isomorphic to G(32, 7), namely
N1 := 〈x, z, w, v, u〉, N2 := 〈xy, z, w, v, u〉.
• The group G(64, 35) contains exactly two subgroups N3, N4 isomorphic to G(32, 6),
namely
N3 := 〈x, z, w, v, u〉, N4 := 〈xy, z, w, v, u〉.
• The group G(64, 37) contains exactly two subgroups N5, N6 isomorphic to G(32, 8),
namely
N5 := 〈x, z, w, v, u〉, N6 := 〈xy, z, w, v, u〉.
In addition, for every i ∈ {1, . . . , 6} we have
(a) [Ni, Ni] = 〈v, u〉 ∼= Z2 × Z2.
(b) y /∈ Ni and y commutes with all elements in [Ni, Ni].
Proof. See the GAP4 script 7 in the Appendix. �
Proposition 4.10. The cases G◦ = G(32, 6), G(32, 7), G(32, 8) do not occur.
Proof. By Lemma 4.6 and Computational Fact 4.9 it follows that, given any nonsplit extension
of type (6) with G◦ as above, condition (M1) in Proposition 2.6 cannot be satisfied. �
Summing up, we finally obtain
Proof of Proposition 4.3. It follows from Propositions 4.8 and 4.10.
5. Moduli spaces
Let Ma,b be the moduli space of smooth minimal surfaces of general type with χ(OS) =
a, K2S = b; by an important result of Gieseker, Ma,b is a quasiprojective variety for all a, b ∈ N
(see [Gie77]). Obviously, our surfaces are contained in M1,8 and we want to describe their locus
there. We denote by M(G,m,n) the moduli space of unmixed surfaces of type (G,m,n) and by
M(G,n) the moduli space of mixed surfaces of type (G,n). We know that n = (22), (3) or (2) in
the unmixed case, whereas n = (22) in the mixed one. By a general result of Catanese ([Ca00]),
both M(G,m,n) and M(G,n) consist of finitely many irreducible connected components of
M1,8, all of the same dimension. More precisely, we have
dim M(G,m,n) = r + s− 3, dim M(G,n) = s.
Consider the mapping class groups in genus zero and one:
Mod0,[r] := 〈σ1, . . . , σr | σiσi+1σi = σi+1σiσi+1,
σiσj = σjσi if |i− j| ≥ 2,
σr−1σr−2 · · · σ
1 · · · σr−2σr−1 = 1〉,
Mod1,1 := 〈tα, tβ , tγ | tαtβtα = tβtαtβ, (tαtβ)
3 = 1〉,
Mod1,[2] := 〈tα, tβ , tγ , ρ | tαtβtα = tβtαtβ, tαtγtα = tγtαtγ ,
tβtγ = tγtβ, (tαtβtγ)
4 = 1,
tαρ = ρtα, tβρ = ρtβ, tγρ = ρtγ〉.
One can prove that
Mod0,[r] : = π0 Diff
+(P1 − {p1, . . . , pr}),
Mod1,1 : = π0 Diff
+(Σ1 − {p}),
Mod1,[2] : = π0 Diff
+(Σ1 − {p, q}),
where Σ1 is the torus S
1×S1 ([Schn03], [CattMu04]). This implies that we can define actions of
these groups on the set of generating vectors for G of type (0 | m1, . . . ,mr), (1 | n) and (1 | n
respectively.
If V := {g1, . . . , gr} is of type (0 | m1, . . . ,mr) then the action is given by
gi −→ gi+1
gi+1 −→ g
i+1gigi+1
gj −→ gj if j 6= i, i+ 1.
If W := {ℓ1; h1, h2} is of type (1 | n) then
ℓ1 −→ ℓ1
h1 −→ h1
h2 −→ h2h1
ℓ1 −→ ℓ1
h1 −→ h1h
h2 −→ h2.
If W := {ℓ1, ℓ2; h1, h2} is of type (1 | n
2) then
ℓ1 −→ ℓ1
ℓ2 −→ ℓ2
h1 −→ h1
h2 −→ h2h1
ℓ1 −→ ℓ1
ℓ2 −→ ℓ2
h1 −→ h1h
h2 −→ h2
ℓ1 −→ ℓ1
ℓ2 −→ h1h
1 ℓ2h1h2h
h1 −→ h
2 ℓ1h1
h2 −→ h2
ℓ1 −→ h
1 ℓ2h1h2
ℓ2 −→ h
2 ℓ1h2h1
h1 −→ h
h2 −→ h
These are called Hurwitz moves and the induced equivalence relation on generating vectors is
said Hurwitz equivalence (see [BaCa03], [BaCaGr06], [Pol07]).
Now let B(G, m, n) be the set of pairs of generating vectors (V, W) such that the assumptions
in Proposition 2.2 are satisfied; then we denote by R the equivalence relation on B(G, m, n)
generated by Hurwitz moves on V, Hurwitz moves on W and the simultaneous action of Aut(G)
on V and W. Similarly, let B(G, n) be the set of generating vectors V such that the assump-
tions of Proposition 2.6 are satisfied; then we denote by R the equivalence relation on B(G, n)
generated by the Hurwitz moves and the action of Aut(G) on V.
Proposition 5.1. The number of irreducible components in M(G, m, n) equals the number
of R-classes in B(G, m, n). Analogously, the number of irreducible components in M(G, n)
equals the number of R-classes in B(G, n).
Proof. We can repeat exactly the same argument used in [BaCaGr06, Propositions 5.2 and 5.5];
we must just replace, where it is necessary, the mapping class group of P1 with the mapping
class group of the elliptic curve E. �
Proposition 5.1 in principle allows us to compute the number of connected components of the
moduli space in each case. In practice, this task may be too hard to be achieved by hand, but it
is not out of reach if one uses the computer. Recently, M. Penegini and S. Rollenske developed a
GAP4 script that solves this problem in a rather short time. We put the result of their calculations
in the Main Theorem (see Introduction), referring the reader to the forthcoming paper [Pe08]
for further details.
6. Appendix
In this Appendix we include, for the reader’s convenience, some of the GAP4 scripts that we
have used in our computations; all the others are similar and can be easily obtained modifying
the ones below.
Let us show how the procedure in the proof of Proposition 3.2 applies to an explicit exam-
ple, namely mα(m) = (2, 4, 12)12 . First we find all the nonabelian groups of order 24 that are
(0 | 2, 4, 12)-generated. This is done using GAP4 as below; the output tells us that there is only
one such a group, namely G = G(24, 5).
gap> # -------------- SCRIPT 1 ------------------
gap> s:=NumberSmallGroups(24);; set:=[1..s];
[1..15]
gap> for t in set do
> c:=0; G:= SmallGroup(24,t);
> Ab:=IsAbelian(G);
> for g1 in G do
> for g2 in G do
> g3:=(g1*g2)^-1;
> H:= Subgroup(G, [g1,g2]);
> if Order(g1)=2 and Order(g2)=4 and Order(g3)=12 and
> Order(H)=Order(G) and
> Ab=false then
> c:=c+1; fi;
> if Order(g1)=2 and Order(g2)=4 and Order(g3)=12 and
> Order(H)=Order(G) and
> Ab=false and c=1 then
> Print(IdSmallGroup(G)," ");
> fi; od; od; od; Print("\n");
[24,5]
By using the two instructions P:=PresentationViaCosetTable(G) and TzPrintRelators(P)
we see that G has the presentation 〈x, y | x2 = y12 = 1, xyx−1 = y5〉, hence it is isomorphic to
the metacyclic group D2,12,5.
In order to speed up further computations, we define the sets G2, G4 given by the elements of
G having order 2 and 4, respectively.
gap> G:=SmallGroup(24,5);;
gap> G2:=[];; G4:=[];;
gap> for g in G do
> if Order(g)=2 then Add(G2,g); fi;
> if Order(g)=4 then Add(G4,g); fi; od;
Then we check whether G is actually (1 | 22)-generated; if not, it should be excluded.
gap> c:=0;;
gap> for l2 in G2 do
> for h1 in G do
> for h2 in G do
> l1:=(l2*h1*h2*h1^-1*h2^-1)^-1;
> K:=Subgroup(G, [l2, h1, h2]);
> if Order(l1)=2 and Order(K)=Order(G) then
> Print(IdSmallGroup(G), " is (1 | 2,2)-generated", "\n"); c:=1; fi;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
[24,5] is (1 | 2,2)-generated
We finish the proof by checking whether the surface S actually exists; the procedure is to look
for a pair (V, W) of generating vectors for G satisfying the assumptions of Proposition 2.2.
gap> c:=0;;
gap> for g1 in G2 do
> for g2 in G4 do
> g3:=(g1*g2)^-1;
> H:=Subgroup(G, [g1, g2]);
> for l2 in G2 do
> for h1 in G do
> for h2 in G do
> l1:=(l2*h1*h2*h1^-1*h2^-1)^-1;
> K:=Subgroup(G, [l2, h1, h2]);
> Boole1:=l1 in ConjugacyClass(G, g1);
> Boole2:=l1 in ConjugacyClass(G, g2^2);
> Boole3:=l1 in ConjugacyClass(G, g3^6);
> Boole4:=l2 in ConjugacyClass(G, g1);
> Boole5:=l2 in ConjugacyClass(G, g2^2);
> Boole6:=l2 in ConjugacyClass(G, g3^6);
> if Order(g3)=12 and Order(l1)=2 and
> Order(H)=Order(G) and Order(K)=Order(G) and
> Boole1=false and Boole2=false and Boole3=false and
> Boole4=false and Boole5=false and Boole6=false then
> Print("The surface exists "); c:=1; fi;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od;
> if c=1 then break; fi; od; Print("\n");
The surface exists
The script above can be easily modified in order to obtain the list of all admissible pairs (V, W);
for instance, one of such pairs is given by
g1 = x, g2 = xy
−1, g3 = y
ℓ1 = xy
2, ℓ2 = xy
2, h1 = y, h2 = y.
Finally, here are the GAP4 scripts used in Section 4.
gap> # -------------- SCRIPT 2 ------------------
gap> s:=NumberSmallGroups(36);; set:=[1..s];
[1..14]
gap> for t in set do
> c:=0; G:=SmallGroup(36,t);
> N:=NormalSubgroups(G);
> for G0 in N do
> if IdSmallGroup(G0)=[18,3] then
> c:=c+1; fi;
> if IdSmallGroup(G0)=[18,3] and c=1 then
> Print(IdSmallGroup(G), " ");
> fi; od; od; Print("\n");
[36,10] [36,12]
gap> # -------------- SCRIPT 3 ------------------
gap> set:=[2,4,5,6,7,8,12,17];;
gap> for t in set do
> n2:=0;
> G0:=SmallGroup(32,t);
> for g in G0 do
> if Order(g)=2 then
> n2:=n2+1; fi; od;
> Print(IdSmallGroup(G0), " "); Print(n2, " ");
> od; Print("\n");
[32,2] 7 [32,4] 3 [32,5] 7 [32,6] 11 [32,7] 11
[32,8] 3 [32,12] 3 [32,17] 3
gap> # -------------- SCRIPT 4 ------------------
gap> s:=NumberSmallGroups(64);; set:=[1..s];
[1..267]
gap> for t in set do
> c:=0; G:=SmallGroup(64,t);
> N:=NormalSubgroups(G);
> for G0 in N do
> if IdSmallGroup(G0)=[32,2] then
> c:=c+1; fi;
> if IdSmallGroup(G0)=[32,2] and c=1 then
> Print(IdSmallGroup(G), " ");
> fi; od; od; Print("\n");
[64,8] [64,9] [64,56] [64,57] [64,58] [64,59] [64,61] [64,62] [64,63] [64,64]
[64,66] [64,67] [64,68] [64,69] [64,70] [64,72] [64,73] [64,74] [64,75] [64,76]
[64,77] [64,78] [64,79] [64,80] [64,81] [64,82]
gap> # -------------- SCRIPT 5 ------------------
gap> set:=[8,9,56,57,58,59,61,62,63,64,66,67,68,69,70,
>72,73,74,75,76,77,78,79,80,81,82];;
gap> for t in set do
> n2:=0; G:=SmallGroup(64,t);
> for g in G do
> if Order(g)=2 then n2:=n2+1;
> fi; od;
> if n2=7 then
> Print(IdSmallGroup(G), " ");
> fi; od; Print("\n");
[64,9] [64,57] [64,59] [64,63] [64,64] [64,68] [64,70]
[64,72] [64,76] [64,79] [64,81] [64,82]
gap> # -------------- SCRIPT 6 ------------------
gap> set:=[5,7,9,11,13,,14,15,16,28,33,35,37,43,45,46,
>57,59,63,64,68,70,72,76,79,81,82,112,113,114, 122,126,
>127,132,143,156,158,160,164,165,166,172,182];;
gap> for t in set do
> c:=0; G:=SmallGroup(64,t);
> D:=DerivedSubgroup(G);
> for d in D do
> B:=d in Center(G);
> if Order(d)=2 and B=false then
> c:=c+1; fi;
> if Order(d)=2 and B=false and c=1 then
> Print(IdSmallGroup(G), " ");
> fi; od; od; Print("\n");
[64,5] [64,33] [64,35] [64,37]
gap> # -------------- SCRIPT 7 ------------------
gap> s:=[33, 35, 37];; I:=[1, 2, 3];;
gap> r:=[ [[32,6], [32,7]], [[32,6]], [[32,8]] ];;
> for i in I do
> G:=SmallGroup(64, s[i]); Print(IdSmallGroup(G), "\n");
> for N in NormalSubgroups(G) do
> if IdSmallGroup(N) in r[i] then
> Print(N, "="); Print(IdSmallGroup(N), " ");
> Print(DerivedSubgroup(N), "\n");
> fi; od; Print("\n"); od;
[64,33]
Group( [ f1*f2, f3, f4, f5, f6 ] )=[32,7] Group( [ f5, f6 ] )
Group( [ f1, f3, f4, f5, f6 ] )=[32,6] Group( [ f5, f6 ] )
[64,35]
Group( [ f1*f2, f3, f4, f5, f6 ] )=[32,6] Group( [ f5, f6 ] )
Group( [ f1, f3, f4, f5, f6 ] )=[32,6] Group( [ f5, f6 ] )
[64,37]
Group( [ f1*f2, f3, f4, f5, f6 ] )=[32,8] Group( [ f5, f6 ] )
Group( [ f1, f3, f4, f5, f6 ] )=[32,8] Group( [ f5, f6 ] )
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Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121
Padova, Italy.
E-mail address: carnoval@math.unipd.it
Dipartimento di Matematica, Università della Calabria, Via Pietro Bucci, 87036 Arcavacata di
Rende (CS), Italy.
E-mail address: polizzi@mat.unical.it
	0. Introduction
	1. Basic on surfaces isogenous to a product
	2. The case pg=q=1. Building data
	2.1. Unmixed case
	2.2. Mixed case
	3. The unmixed case
	4. The mixed case
	4.1. The case g(C)=5,  |G|=16
	4.2. The case g(C)=7,  |G|=36
	4.3. The case g(C)=9,  |G|=64
	5. Moduli spaces
	6. Appendix
	References
