Non-classical Godeaux Surfaces

Abstract

A non-classical Godeaux surface is a minimal surface of general type with $\chi =K^2=1$ but with $h^{01}\ne 0$. 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.

13 pages, some minor mistakes corrected

References in corpus (1)

• Algebraic Surfaces of General Type with Small c_1^2 in Positive Characteristic

Cited by in corpus (1)

• A note on non-reduced Picard schemes