Semistable degeneration of Godeaux surfaces with relatively nef canonical bundle

Abstract

We classify the central fiber of a semistable degeneration (in the sense of the semistable reduction theorem) of Godeaux surfaces, under the condition that the dualizing sheaf is relatively nef. We prove in particular that if the central fiber is reducible then all its components have negative Kodaira dimension. As a corollary, this result provides a classification of possible central fibers of a semistable degeneratioI having two components.