Tautological families of cyclic covers of projective spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913438197874688 |
|---|---|
| author | Kundu, Promit Mukherjee, Jayan Raychaudhury, Debaditya |
| author_facet | Kundu, Promit Mukherjee, Jayan Raychaudhury, Debaditya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_06043 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Tautological families of cyclic covers of projective spaces Kundu, Promit Mukherjee, Jayan Raychaudhury, Debaditya Algebraic Geometry 14A20, 14C15, 14D22, 14D23, 14E08, 14F22 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. |
| title | Tautological families of cyclic covers of projective spaces |
| topic | Algebraic Geometry 14A20, 14C15, 14D22, 14D23, 14E08, 14F22 |
| url | https://arxiv.org/abs/2111.06043 |