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.