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In this article, we study the existence of tautological families on a Zariskiopen set of the coarse moduli space parametrizing certain Galois covers overprojective spaces. More specifically, let ($1$) $\mathscr{H}_{n.r.d}$ (resp.$M_{n,r,d}$) be the stack (resp. coarse moduli) parametrizing smooth simplecyclic covers of degree $r$ over the projective space $\mathbb{P}^n$ branchedalong a divisor of degree $rd \geq 4$, and ($2$) $\mathscr{H}_{1,3,d_1,d_2}$(resp. $M_{1,3,d_1,d_2}$) be the stack (resp. coarse moduli) of smooth cyclictriple covers over $\mathbb{P}^1$ with $2d_1-d_2 \geq 4$ and $2d_2-d_1 \geq 4$.In the former case, we show that such a family exists if and only if$\textrm{gcd}(rd, n+1) \mid d$ while in the latter case, we show that it alwaysexists. We further show that even when such a family exists, often it cannot beextended to the open locus of objects without extra automorphisms. Theexistence of tautological families on a Zariski open set of its coarse modulican be interpreted in terms of rationality of the stack if the coarse modulispace is rational. Combining our results with known results on the rationalityof the coarse moduli of points on $\mathbb{P}^1$ and the coarse moduli of planecurves, we determine the rationality of $\mathscr{H}_{1,r,d}$ (resp.$\mathscr{H}_{2,r,d}$) for $rd \geq 4$ (resp. $rd\geq 49$). On the other hand$\mathscr{H}_{1,3,d_{1},d_{2}}$ is unirational, and we show that its coarsemoduli $M_{1,3,d_1,d_2}$ is unirational and fibred over a rational base byhomogeneous varieties which are rational if $\textrm{char}(\mathbb{k}) = 0$.Our study is motivated by the work of Gorchinskiy and Viviani on the moduli ofhyperelliptic curves.

Tautological families of cyclic covers of projective spaces