Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tautological families of cyclic covers of projective spaces   In this article, we study the existence of tautological families on a Zariski
open set of the coarse moduli space parametrizing certain Galois covers over
projective spaces. More specifically, let ($1$) $\mathscr{H}_{n.r.d}$ (resp.
$M_{n,r,d}$) be the stack (resp. coarse moduli) parametrizing smooth simple
cyclic covers of degree $r$ over the projective space $\mathbb{P}^n$ branched
along 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 cyclic
triple 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 always
exists. We further show that even when such a family exists, often it cannot be
extended to the open locus of objects without extra automorphisms. The
existence of tautological families on a Zariski open set of its coarse moduli
can be interpreted in terms of rationality of the stack if the coarse moduli
space is rational. Combining our results with known results on the rationality
of the coarse moduli of points on $\mathbb{P}^1$ and the coarse moduli of plane
curves, 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 coarse
moduli $M_{1,3,d_1,d_2}$ is unirational and fibred over a rational base by
homogeneous varieties which are rational if $\textrm{char}(\mathbb{k}) = 0$.
Our study is motivated by the work of Gorchinskiy and Viviani on the moduli of
hyperelliptic curves.

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The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin