Factoriality of normal projective varieties

For a normal projective variety $X$, the $\mathbf{Q}$-factoriality defect $\sigma (X)$ is defined to be the rank of the quotient of the group of Weil divisors by the subgroup of Cartier ones. We prove a slight improvement of a topological formula of S.G. Park and M. Popa asserting that $\sigma (X)=h^{2n-2}(X)-h^2(X)$ by assuming only 2-semi-rationality, that is, $R^k{\pi }_{\ast }{\mathcal{O}}_{\tilde{X}}=0$ for $k=1,2$, instead of rational singularities for $X$, where $\pi :\tilde{X}\to X$ is a desingularization with $h^k(X):=\dim H^k(X,\mathbf{Q})$ and $n:=\dim X>2$. Our proof generalizes the one by Y. Namikawa and J.H.M. Steenbrink for the case $n=3$ with isolated hypersurface singularities. We also give a proof of (a slight generalization of) the assertion that $\mathbf{Q}$-factoriality implies factoriality if $X$ is a local complete intersection whose singular locus has at least codimension three. These imply a slight improvement of Grothendieck’s theorem in the projective case asserting that $X$ is factorial if it is a local complete intersection whose singular locus has at least codimension three and at general points of its components of codimension three, $X$ has rational singularities and is a $\mathbf{Q}$-homology manifold.