$\mathbb Z/2$-Godeaux surfaces

Abstract

We compute explicit equations for all (universal coverings of) Godeaux surfaces with torsion group Z/2. We show that their moduli space is irreducible and rational of dimension 8. 2020 MSC: 14J29[SS20] claiming the construction of an 8-dimensional family of simply connected Godeaux surfaces, but without obtaining a full classification).Catanese and Debarre [CD89] showed that the étale double covers of Z/2-Godeaux surfaces have hyperelliptic canonical curve and birational bicanonical map onto an octic in P 3 , and they did a general study of its canonical ring. That octic is given by the determinant of a certain matrix α.In this paper we continue their work. Using an idea from Miles Reid [Rei90], we get more precise information about α by looking first to its restriction to the case of the canonical curve, then extending to the surface. Then we give an algorithm for the computation of all such matrices, from which we obtain equations for the étale double covers of all Z/2-Godeaux surfaces. We show that their moduli space is irreducible of dimension 8, which implies that the topological fundamental group of Z/2-Godeaux surfaces is also Z/2.We note that our method is not brute force computation: for the main algorithm, the calculations used only 32 MB of RAM memory, and took 85 seconds on a low-end computer.Recently two special Z/2-Godeaux surfaces have appeared in the literature: a (Z/3) 2 -quotient of a fake projective plane, constructed by Borisov and Fatighenti [BF20], which has 4 cusp singularities; a degree 6 quotient of the so-called Cartwright-Steger surface, given by Borisov-Yeung [BY20], which has 3 cusp singularities and a certain configuration of rational curves. As an exercise, we give the coordinates of these surfaces in our family.All computations are implemented with Magma [BCP97], and can be found in some arXiv ancillary files. In particular, using the files 5_Verifications_alpha_i_c_j.txt one can choose any surface in the family and compute its invariants and singular set. Results from Catanese-DebarreWe collect here some results from the paper [CD89] that will be used throughout the text.Let S be the étale double cover of a numerical Godeaux surface with torsion group Z/2, and denote the corresponding involution by σ. The invariants of S are K 2 = 2, p g = 1, q = 0. Define the canonical ring of S as R = ∞ n=0 H 0 (S, nK S ), and let A = C[x, y 1 , y 2 , y 3 ] be the C-graded algebra with deg(x) = 1, deg(y i ) = 2.

Rational configurations in K3 surfaces and simply-connected $p_g=1$ surfaces for $K^2=1,2,3,4,5,6,7,8,9$

“…If that were the case, then all these surfaces would be simply-connected. We note that these particular surfaces are relevant for the description of the moduli space of Z/2-Godeaux surfaces, which was recently proved [DR20] to be a rational irreducible variety of dimension 8, finishing with the previous attempts e.g. [CD80,CCO94,Co16].…”

Smoothing semi-smooth Stable Godeaux surfaces

“…As we recalled above, the algebraic fundamental group π alg 1 of a Godeaux surface is cyclic of order m ≤ 5; it is a folklore conjecture that for each value of m ≤ 5 the connected component of the moduli of Godeaux surfaces with π alg 1 = Z m is irreducible and rational of dimension 8. This conjecture is known to be true for m ≥ 3 by [Rei76] (see also [CU18]) and for m = 2 by the recent preprint [DR20].…”

On degenerations of $\mathbb{Z}/2$-Godeaux surfaces

“…[19], respectively. (In the recent preprint [9], the first two authors have settled Reid's conjecture for Z=2-Godeaux surfaces. The starting point of [9] relies on the computations done in this paper.)…”

“…(In the recent preprint [9], the first two authors have settled Reid's conjecture for Z=2-Godeaux surfaces. The starting point of [9] relies on the computations done in this paper.) Several authors have worked on these surfaces, and there are some unrelated constructions of some components of the moduli space.…”