Suppose $X$ is a projective variety, and $D$ is a Cartier divisor on $X$. Is it possible (or this is always true) that there is a (ramified) double cover $\pi: \tilde{X} \to X$ such that $\frac{1}{2}\pi^\*(D)$ is also a Cartier divisor?



Cartier divisor on a double cover
This is not always possible. Take $X=\mathbb{P}^2$, $D=$ a line. If $\pi ^\*D$ is twice a Cartier divisor on $\tilde{X} $, this holds also on the normalization of $\tilde{X} $, so you can assume that $\tilde{X} $ is normal. Then $(\frac{1}{2}\pi ^\*D)^2 = \frac{1}{2}D^2=\frac{1}{2}$; but on a normal surface the intersection of two Cartier divisors is an integer.


