Factoriality of normal projective varieties

Abstract

For a normal projective variety $X$, the $\mathbb{Q}$-factoriality defect $\ddot{I}ƒ(X)$ is defined to be the rank of the quotient of the group of Weil divisors by the subgroup of Cartier ones. We prove an improvement of a topological formula of S.G. Park and M. Popa asserting that $\ddot{I}ƒ(X)\le h^{2n-2}(X)-h^2(X)$ by assuming only 1-semi-rational singularities instead of rational singularities, and the equality holds in the 2-semi-rational case. Here the singularities are called $j$-semi-rational if $R^k\ddot{I}{€}_{\ast }{\mathcal{O}}_{\tilde{X}}=0$ for any $k\in [1,j]$ with $\ddot{I}€:\tilde{X}\to X$ a desingularization, $h^k(X):=\dim H^k(X,\mathbb{Q})$, and $n:=\dim X\ge 2$. We also show (a slight generalization of) the assertion that $\mathbb{Q}$-factoriality implies factoriality if $X$ is a local complete intersection whose singular locus has at least codimension three. We then get a new proof for the projective case of Grothendieck's theorem asserting that $X$ is factorial if it is a local complete intersection whose singular locus has at least codimension four. We also show that a local complete intersection $X$ of dimension 3 having only isolated singularities is factorial if the (topological) defect $\mathrm{def}(X):=h^4(X)-h^2(X)$ vanishes, without any assumption on rational singularities.

This is an extended version of arXiv:2512.23522v1, Section 4