# Does pull-push by the quotient map of divisors on the symmetric square of an algebraic curve induce multiplication-by-2?

Let $C$ be a smooth projective algebraic curve over a field $k$ of characteristic different from 2, and let $C^2 = C \times C$ be the square of $C$. Let $C^{(2)} = \operatorname{Sym}^2(C)$ be the symmetric square of $C$; that is, $C^{(2)}$ is the quotient of $C^2$ by the group $G = \{1, \sigma\}$ where $\sigma\colon C^2 \to C^2$ is defined by $\sigma(P,Q) = (Q,P)$. Write $\pi\colon C^2 \to C^{(2)}$ for the quotient map.

Question: Is the induced map $\pi_*\pi^*\colon \operatorname{Div}(C^{(2)}) \to \operatorname{Div}(C^{(2)})$ equal to "multiplication-by-2" on $\operatorname{Div}(C^{(2)})$? If so, is there an easy way to prove it? In any case, what is a good reference for this kind of question?

It seems to me that this ought to be true by some analogue of the Riemann-Hurwitz formula for surfaces, but I haven't had any luck finding such a result.

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This is true in great generality. If $f: X\to Y$ is a finite surjective morphism of degree $d$ between regular noetherian schemes, then for any Weil divisor $D$ on $Y$, we have $$f_*f^*D=dD.$$ (The regularity hypothesis can be merely replaced by integrality if $Y$ is finite type over a field, the only issue is the dimension of $f(D)$ in general).
In your case, $\pi$ has degree $2$.