1. Introduction
Understanding moduli spaces is an important step for completing the classification process in algebraic geometry, a problem which is not yet settled, even in dimension 2. One of the interesting classes in dimension 2 consists of the surfaces of general type. In [
1], Gieseker proved the existence of the modulus varieties
, parameterising the surfaces of general type with given invariants
and
. The ultimate goal is to understand its compactification
—the moduli space of stable surfaces that, intuitively, are the limits of smooth surfaces. One of the basic questions about the geometry of
is to find and analyze the codimension 1 components of the boundary. One important boundary divisor consists of normal surfaces with Wahl singularities, also known as T-singularities [
2].
In this work, we are interested in studying the moduli space of Campedelli surfaces, which are the surfaces of general type with
and
. Campedelli surfaces are the first example of non-rational surfaces with
[
3]. Every finite group of order ≤9 can occur as the fundamental group for such surfaces, except for Dihedral groups of order 6 and 8. We consider only the Campedelli surfaces with a fundamental group of order 8 [
4].
Two minimal surfaces of general type belong to the same connected component of the Gieseker moduli scheme iff they are deformation equivalent. Moreover, surfaces in the same connected component have the same fundamental group. We aim to find two deformations of the Godeaux surface with
such that the central fibre contains the resolution graph of a T-singularity. Using unprojection, we then show that there exist two connected components at the boundary of
that contain surfaces birational to Godeaux surfaces. In [
5], Alexeev and Pardini showed that, in the compactification of the moduli space of the Campedelli surfaces with
, Godeaux surface
T with
does not appear. There are three Campedelli surfaces corresponding to Abelian fundamental groups of order 8. In this work, we consider the other two cases. There are Campedelli surfaces with non-Abelian fundamental groups of order 8, which are not considered.
The key idea in the present work is derived from [
6] (Lemma 2.3) which describes how different T-singularities change
in the case of canonically embedded algebraic surfaces. Based on this, there are two possible directions. The forward part is to consider the
-Gorenstein deformations of a surface such that the central fibre contains a desired T-singularity, then resolving the central fibre decrease
by a specified positive integer. The reverse part is to deform a surface so that it contains the resolution graph of a desired T-singularity, contracting the resolution graph increase
. In [
6], we adapted the forward way and constructed some deformations of Campedelli surfaces with two different fundamental groups to contain
singularities and, thus, proved the existence of the surfaces birational to the Godeaux surface with
in two different connected components of
. In this work, we consider the reverse part, that is, constructing the deformations of the Godeaux surface with
to contain the resolution graphs of
. For the canonically embedded surfaces, these resolution graphs are conics. As a consequence of our deformation construction, we are able to construct Campedelli surfaces, with higher
, from a Godeaux surface, a surface with a low value of
. In general, the reverse part is important since the construction of these surfaces of general type is harder for higher values of
.
One of the main challenges is that the deformation spaces of such surfaces of general type satisfy Murphy’s law and can be arbitrarily singular [
7]. We aim to find the deformations of the universal cover of a Godeaux surface
T such that each central fibre contains specific resolution graphs without any additional singularities. We use the group action on the canonical ring of the universal cover of the Godeaux surface
T to achieve our goal. Here is the main result which addresses this issue.
Theorem 1. There exist two deformations of Godeaux surface T with fundamental group , such that each central fibre contains a conic.
To contract these conics, we use unprojection. To unproject these conics and to find the fundamental group of the unprojections, we use key varieties constructed using Triple Tom & Jerry [
8], an unprojection format introduced by Reid et al. These key varieties are sixfold and make our constructions explicit in the sense that the generators and relations of polarised graded rings are given at each stage.
In
Section 2, we recall the basics of unprojection and Tom & Jerry, a format for unprojection.
Section 3 is our main section, where we use group action on the graded rings to find two deformation families of a sixfold. Using the central fibres of these deformations and the theory of unprojection, we construct our key varieties in
Section 4. In
Section 5, we first review the construction of the Godeaux surfaces
T with fundamental group
from [
9], and then construct two families inside
using previous constructions. We then construct stable surfaces with
singularities in
Section 6. Using these families and
-Gorenstein smoothing, introduced by Lee and Park in this context [
10,
11], we construct Campedelli surfaces [
12,
13] with fundamental groups
and
.
Notation: We work over the field of complex numbers .
6. Constructing Campedelli Surfaces from Godeaux with
A Campedelli surface is a surface of general type with
, and the order of the algebraic fundamental group
is at most 9 [
21]. Our focus is on Campedelli surfaces with
.
Theorem 4 ([
12]).
Let X be a Campedelli surface, and be an etale cover of degree 8. The canonical model of Z is isomorphic to a complete intersection of four quadrics . Moreover, Z is the universal cover of X and the covering group is the topological fundamental group . We denote by the Campedelli surfaces with fundamental groups and . Let be the etale Galois cover for , such that .
Lemma 5. Unprojecting , for , we obtain surfaces with and containing singularities with a action. The singularities form an orbit under the action.
Proof. The surfaces
(
35) have invariants
,
, containing four conics
(
36) permuted by
action (
37). Since
and
the unprojection
can be obtained by using the unprojection of
given in (
22). Using key varieties
, the surfaces after the unprojection of all four conics are given by
Using [
6] (Lemma 2.3), we obtain
and
. The four singularities form an orbit because of the action of
on the unprojection variables given in (
25). □
Theorem 5. There exist -Gorenstein smoothings of such that the fibres are invariant under the respective group actions of .
Proof. To obtain a
-Gorenstein smoothing of
, we move the quadric sections (
34); the new quadrics are given as
where
are parameters. It can be seen that
and, hence, the general fibre, given below, is invariant under the action of
:
The coordinate ring of the general fibre is given by
where
,
, and
E are
minors of Matrix (
28).
The sixfold is obtained through unprojection from , which is Gorenstein since it is a complete intersection. Also, an unprojection of a Gorenstein is a Gorenstein. Hence, is Gorenstein, being a complete intersection with a Gorenstein. □
Theorem 6. The fundamental group of the general fibre is . Moreover, there exists an etale Galois cover of given as an intersection of four quadrics in .
Proof. For
, the key variety
is constructed in terms of the general elements of
and
. We can take
with orbits
and
, such that
Let
be the square matrix of the coefficients of
with respect to variables
. The
are non-singular so we can express
in terms of the symbols
, and use these as our new variables. Similarly, for
, the symbols become
.
Let
, for
, be the quadrics in terms of
. Then,
where
E is given as the
minors of the following matrix:
Thus,
where
are the quadrics in our new variables. Let
be the surface defined by these four quadrics. The surfaces
are smooth by Bertini’s theorem. The action of
on
is given by
Moreover, the group
acts on
as (
5) and, from the matrix
, we have
□
Corollary 1. The surface , together with the action of , gives a construction of Campedelli with .
Proof. The action of
on
is obtained by combining the action of
given in (
51) and of
given as
This gives a construction of Campedelli with fundamental group
(c.f. page 13 in [
13]). The situation is similar for
. □
7. Conclusions
We constructed two deformations of the Godeaux surface with , each containing a family of conics in the central fiber. By using -Gorenstein smoothing and unprojection techniques, we linked families of surfaces birational to these Godeaux surface to two distinct components of the Campedelli surfaces with fundamental group of order 8.
These deformations provide insight into the birational geometry of Godeaux and Campedelli surfaces. While our study focuses on explicit constructions using deformation and unprojection methods, our results may have broader implications for the study of algebraic cycles and cohomological invariants, potentially offering new examples relevant to motivic cohomology.
Additionally, our work explores the structure of polynomial rings under the action of the fundamental group. By using graded rings with group actions, we provide explicit constructions. This provides a new perspective on the interplay between group actions and the study of varieties.