Nonrationality of nodal quartic threefolds (Q388797)

The paper under review proves the factoriality of quartic 3-folds with 13 nodes as only singularities if no plane or quadrics are contained. The factoriality of hypersurfaces is an interesting problem also due to its connection with rationality [\textit{M. Mella}, Math. Ann. 330, No. 1, 107--126 (2004; Zbl 1058.14022)]. In recent years many results, [\textit{I. Cheltsov}, J. Algebr. Geom. 14, No. 4, 663--690 (2005; Zbl 1084.14039)], has been proved but a full picture is still missing. In this note the result in [\textit{I. Cheltsov}, Pac. J. Math. 226, No. 1, 65--81 (2006; Zbl 1123.14010)] is improved via nice arguments of synthetic geometry.