Factoriality condition of some nodal threefolds in \({\mathbb{P}^4}\) (Q958177)

In the paper under review, the author studies the problem of factoriality of a nodal hypersurface \(X\) of degree \(n\) in \(\mathbb{P}^4\) (``factoriality'' means that every Weil divisor on \(X\) is Cartier). \textit{I. Cheltsov} [J. Algebr. Geom. 14, No. 4, 663--690 (2005; Zbl 1084.14039)] proved that \(X\) is factorial if the singular locus of \(X\) consists of at most \(\displaystyle\frac{1}{4}(n - 1)^{2}\) nodes. However, the expected estimate for the number of nodes, under which \(X\) will be factorial, is \((n - 1)^2 - 1\), as was conjectured by C. Ciliberto (note that this bound is sharp). As the main result (see Theorem 1.7), it is proved that \(X\) is factorial under the \(\leqslant(n - 1)^2 - 1\) bound for \(n = 8, 9, 10, 11\). The proof employs a necessary condition of factoriality stating that \(X\) is factorial if the singular locus of \(X\) imposes independent linear conditions on homogeneous forms of degree \(2n - 5\) [see \textit{H. Clemens}, Adv. Math. 47, 107--230 (1983; Zbl 0509.14045)]. Note that the conjecture of Ciliberto was completely proved recently by \textit{I. A. Cheltsov} [Sb. Math. 201(7), 1069--1090 (2010; Zbl 1215.14040)].