Definition
A Godeaux surface is a smooth, minimal complex algebraic surface of general type characterized by having geometric genus $p_g = 0$, irregularity $q = 0$, and self‑intersection number of the canonical divisor $K^2 = 1$. It is one of the simplest examples of a surface of general type with the smallest possible values of these invariants.
Overview
Godeaux surfaces were first constructed in the 1930s as examples demonstrating that the invariants $p_g$, $q$, and $K^2$ can take the minimal values compatible with a surface of general type. They are typically obtained as quotients of a higher‑dimensional variety by a finite group action. The classical construction starts with a quintic hypersurface in $\mathbb{P}^3$ (a degree‑5 surface) and takes the quotient by a cyclic group of order five acting freely. The resulting surface is a Godeaux surface. Over the years, several families of Godeaux surfaces have been identified, differing by the choice of the group action and the underlying covering surface.
Etymology / Origin
The surface is named after the Belgian mathematician Lucien Godeaux (1909–2002), who studied surfaces of general type and contributed to the classification of algebraic surfaces. His work in the 1930s provided the first explicit examples of surfaces with $p_g = 0$ and $K^2 = 1$, which later became known as Godeaux surfaces.
Characteristics
| Property | Value / Description |
|---|---|
| Geometric genus ($p_g$) | 0 |
| Irregularity ($q$) | 0 |
| Canonical self‑intersection ($K^2$) | 1 |
| Kodaira dimension | 2 (general type) |
| Fundamental group | Typically a finite cyclic group of order 5, though other groups can occur in different constructions. |
| Construction | Often realized as the quotient of a smooth quintic surface in $\mathbb{P}^3$ by a fixed‑point‑free action of $\mathbb{Z}/5\mathbb{Z}$. Alternative constructions use other cyclic groups or more intricate group actions on higher‑degree hypersurfaces. |
| Hodge numbers | $h^{1,0}=h^{0,1}=0$, $h^{2,0}=h^{0,2}=0$; thus the Hodge diamond is concentrated in the middle rows. |
| Pluricanonical maps | The bicanonical map of a Godeaux surface is generally not birational; the canonical linear system $ |
| Moduli | Godeaux surfaces form a 5‑dimensional component of the moduli space of surfaces of general type with the given invariants. The moduli space is not unirational, and the generic member has torsion in its Picard group. |
Related Topics
- Surfaces of General Type – the broad classification of complex surfaces with maximal Kodaira dimension.
- Barlow Surface – another surface of general type with $p_g = 0$ and $K^2 = 1$, distinct from the Godeaux surface.
- Campedelli Surface – a surface of general type with $p_g = q = 0$ and $K^2 = 2$.
- Fundamental Group of Algebraic Surfaces – study of how finite groups act freely on smooth varieties to produce new surfaces.
- Enriques–Kodaira Classification – the overall scheme categorizing compact complex surfaces according to their Kodaira dimension and other invariants.
These topics provide context for the role of Godeaux surfaces within the broader landscape of algebraic geometry and the classification of complex surfaces.