Rational correspondences between moduli spaces of curves defined by Hurwitz spaces (Q425295)

Let \(C\) be a smooth projective curve of genus \(g=2k\) and let \(\gamma\) be a linear system of degree \(d=k+1\) and projective dimension one. The trace curve associated to the pair \((C,\gamma)\) is defined by \(T_{C,\gamma}=\{(p,q)\in C\times C:\gamma\geq p+q\}\), and is of genus \(g'=5k^2-4k+1\) for a generic \(\gamma\). Its quotient by the involution induced by interchanging the factors of \(C\times C\) is called the reduced trace curve whose genus is given by \(\hat{g}=(5k-2)(k-1)/2\). Since the constructions of these two curves can be done in families, one can define two morphisms \(\phi:H_{2k,k+1}\rightarrow\mathcal{M}_{g'}\) and \(\hat{\phi}:H_{2k,k+1}\rightarrow\mathcal{M}_{\hat{g}}\), where \(H_{g,d}\) is the Hurwitz scheme of simple covers of the projective line of degree \(d\) and genus \(g\), and \(\mathcal{M}_g\) is the moduli space of stable curves of genus \(g\). Let \(p:H_{2k,k+1}\rightarrow\mathcal{M}_{2k}\) denote the natural projection map. The main body of this paper is devoted to calculating the composite maps \(p_{*}\phi^{*}:\mathrm{Pic}(\bar{M}_{g'})\rightarrow \mathrm{Pic}(\bar{M}_g)\) and \(p_{*}\hat{\phi}^{*}:\mathrm{Pic}(\bar{M}_{\hat{g}})\rightarrow \mathrm{Pic}(\bar{M}_g)\) .