Non-classical Godeaux Surfaces

TL;DR

This paper studies non-classical Godeaux surfaces, showing they exist only in positive characteristic up to 5, classifying them in characteristic 5, and providing explicit examples and cohomological computations.

Contribution

It proves that non-classical Godeaux surfaces have $h^{01}=1$, exist only in characteristic at most 5, and offers a complete classification in characteristic 5 with explicit examples.

Findings

Non-classical Godeaux surfaces have $h^{01}=1$.

Such surfaces only exist over fields of characteristic ≤ 5.

Complete classification and explicit example in characteristic 5.

Abstract

A non-classical Godeaux surface is a minimal surface of general type with $\chi=K^2=1$ but with $h^{01}\neq0$. We prove that such surfaces fulfill $h^{01}=1$ and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute their Hodge-, Hodge--Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux surface in characteristic 5.