  Let $C$ be a compact complex curve included in a non-singular complex surface
such that the normal bundle is topologically trivial. Ueda studied complex
analytic properties of a neighborhood of $C$ when $C$ is non-singular or is a
rational curve with a node. We propose an analogue of Ueda's theory for the
case where $C$ admits nodes. As an application, we study singular Hermitian
metrics with semi-positive curvature on the anti-canonical bundle of the
blow-up of the projective plane at nine points in arbitrary position.
