On factorial double solids with simple double points (Q856356)

The article is the complement of the preprint ``Sextic double solids'' by \textit{I. Cheltsov} and \textit{J. Park} [\url{arXiv:math/0404452}]. The authors state the following conjecture (Conjecture 1.4). Let \(S\subset \mathbb P^3\) be a nodal surface of degree \(2r\). Suppose that the surface \(S\) has at most \(r(2r-1)+1\) singular points. Then the double cover of \(\mathbb P^3\) ramified along \(S\) is not \(\mathbb Q\)-factorial (that is, there exists a Weil divisor such that any multiple of it is not Cartier divisor) if and only if the surface \(S\) is defined by the equation of the form \[ f_r(x,y,z,w)^2+h_1(x,y,z,w)g_{2r-1}(x,y,z.w)=0, \] where \(f_r\), \(g_{2r-1}\), \(h_1\) are homogeneous polynomials of degrees \(r\),\(2r-1\), and \(1\), respectively. The authors prove this conjecture for \(r=2\) (Theorem 4.3) and for \(r=3\) (Theorem 5.3). The main instrument of the proofs is the method of H. Clemens, which reduces the computation of the rank of 4-th integral cohomology of double solids to a combinatorial problem.