TITLE: When is an algebraic variety $\mathbb{Q}$-factorial?
QUESTION [12 upvotes]: A variety is $\mathbb{Q}$-factorial if every global Weil divisor is $\mathbb{Q}$-Cartier. How bad singularities are allowed so that the algebraic variety is still $\mathbb{Q}$-factorial? Is a singular curve $\mathbb{Q}$-factorial? For example is a nodal-cuspidal plane curve $\mathbb{Q}$-factorial?

REPLY [9 votes]: A complete intersection $X$ in $\mathbb{P}^n$ is $\mathbb{Q}$ factorial if $\dim X_{sing}<\dim X-3$.
In general being $\mathbb{Q}$-factorial depends on the type of singularity you have, but also on the position of the singularities.
E.g., a degree $d$ threefold $X$ in $\mathbb{P}^4$ with only ordinary double points at $p_1,\dots,p_k$ is $\mathbb{Q}$-factorial if and only if the linear system of homogeneous polynomials of degree $2d-4$ polynomials through the singular points of $X$ has no defect (i.e., the codimension of this linear system is precisely $k$).
An example of a non-$\mathbb{Q}$-factorial threefold with only nodes is $f_1f_2+f_3f_4=0$, where $\deg(f_1)+\deg(f_2)=\deg(f_3)+\deg(f_4)$, and the $f_i$ are chosen sufficiently general.