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We work over $\mathbb{C}$. I'm trying to understand the following result (a lemma from the Stacks project), in some particular example. The Lemma says that for an algebraic stack $X$ for which the inertia stack is quasi-compact over $X$ there exists a stratification by gerbes. See the link for a precise statement and proof.

I am trying to understand what is this decomposition in a simple case: take a neutral gerbe of the form $C/G_m$ where $C$ is, say, a smooth projective curve and $G_m$ is the multiplicative group, and form the 2nd symmetric power, i.e. $(C/G_m\times C/G_m)/\Sigma_2$. Can we describe explicitely a nice stratification by gerbes?

I guess it should be related to $C^{(2)}/G_m^2$ with something happening on the diagonal but I don't really know how...

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