An 8-dimensional family of simply connected Godeaux surfaces. (English) Zbl 1518.14052

Schreyer, Frank-Olaf; Stenger, Isabel

Trans. Am. Math. Soc. 376, No. 5, 3419-3443 (2023).

Smooth, minimal projective surfaces of general type with the smallest numerical invariants, geometric genus \(p_g=0\), and self-intersection of the canonical divisor \(K^2=1\) are called numerical Godeaux surfaces.
By the works of Y. Miyaoka [Invent. Math. 34, 99–111 (1976; Zbl 0337.14010)], and M. Reid [J. Fac. Sci., Univ. Tokyo, Sect. I A 25, 75–92 (1978; Zbl 0399.14025)], we know that the possible torsion of a numerical Godeaux surface are the cyclic groups \(\mathbb{Z}/n\) for \(1\leq n \leq 5\).
For \(n=5,4,3\) in [M. Reid, J. Fac. Sci., Univ. Tokyo, Sect. I A 25, 75–92 (1978; Zbl 0399.14025); S. Coughlan and G. Urzúa, Int. Math. Res. Not. 2018, No. 18, 5609–5637 (2018; Zbl 1444.14071)] is proved that the torsion and the topological fundamental groups coincide. Recently, E. Dias and C. Rito computed equations for numerical Godeaux surfaces with torsion group \(\mathbb{Z}/2\). In all those cases, the moduli space is unirational and irreducible of dimension 8.
For numerical Godeaux surfaces with trivial torsion, it is still open to know if its fundamental topological group is also trivial.
In this paper, the authors found by algebraic-homological and computational methods an \(8\)-dimensional family of numerical Godeaux surfaces with trivial fundamental group.
As the authors remark, some outstanding questions remain open, for example: Do the torsion-free Godeaux surfaces form a single family?

Reviewer: Sergio Troncoso (Valparaíso)

MSC:

14J10	Families, moduli, classification: algebraic theory
14J29	Surfaces of general type
13D02	Syzygies, resolutions, complexes and commutative rings

Keywords:

surfaces of general type; Godeaux surfaces; syzygy computation; free resolution; canonical models

Citations:

Zbl 0337.14010; Zbl 0399.14025; Zbl 1444.14071

Software:

Macaulay2

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