We denote by \Hbar_g the closure of the hyperelliptic locus in the moduli space of stable curves of genus g. We consider the map Sym^2(\Pic(\Hbar_g)) \to CH^2(\Hbar_g) and prove the kernel of the map is generated by a single relation. Moreover, the relation depends on the parity of g, but otherwise the relation has a simple recursive form.

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