We prove that an arbitrary reducible curve on a smooth surface has an essentially unique decomposition into chain-connected curves. Using this decomposition, we give an upper bound of the geometric genus of a numerically Gorenstein surface singularity in terms of certain topological data determined by the canonical cycle. We show also that the fixed part of the canonical linear system of a 1-connected curve is always rational, that is, the first cohomology of its structure sheaf vanishes.
Kazuhiro KONNO. "Chain-connected component decomposition of curves on surfaces." J. Math. Soc. Japan 62 (2) 467 - 486, April, 2010. https://doi.org/10.2969/jmsj/06220467