On the classification of singular cubic threefolds

We classify combinations of isolated singularities that can occur on complex cubic threefolds generalizing analogous results for cubic surfaces (see Schläfli [Philos. Trans. Roy. Soc. 153 (1863), pp. 193–241] and Bruce and Wall [J. London Math. Soc. (2) 19 (1979), pp. 245–256]). In addition, we provide concise combinatorial description of the possible configurations of simple singularities: they essentially correspond to subgraphs of a certain graph.