Exceptional divisor of the Hilbert-Chow morphism of the punctual Hilbert scheme

Let $X$ be a smooth and projective variety of dimension $d>1$. Let $X^{[2]}$ denote the Hilbert scheme of length two subschemes of $X$. Let $X^{(2)}:=Xtimes X/mathbb{Z}_2$, where $mathbb{Z}_2$ acts by $(x,y)mapsto (y,x)$. Then there is a birational map $X^{[2]}to X^{(2)}$. Let $E$ denote the exceptional divisor if this map. Or $E$ can be described as the divisor whose locus is the set of non-reduced subschemes. Can someone point a nice reference where it is explained that there is a line bundle, whose square is the line bundle corresponding to $E$.

Alternatively, the question can be posed as follows. Let $Y$ denote the blow up of the diagonal of $Xtimes X$. Then the action of $mathbb{Z}_2$ extends to $Y$, (I think this action is trivial when restricted to the exceptional divisor). The quotient is $pi:Yto X^{[2]}$. How do I see that there is a divisor $F$ on $X^{[2]}$ such that $2F=pi_*(E)$?

I am looking for a reference or explanation of this fact.