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Tautological families of cyclic covers of projective spaces

Promit Kundu, Jayan Mukherjee, Debaditya Raychaudhury

Abstract

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.

Tautological families of cyclic covers of projective spaces

Abstract

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 () (resp. ) be the stack (resp. coarse moduli) parametrizing smooth simple cyclic covers of degree over the projective space branched along a divisor of degree , and () (resp. ) be the stack (resp. coarse moduli) of smooth cyclic triple covers over with and . In the former case, we show that such a family exists if and only if 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 and the coarse moduli of plane curves, we determine the rationality of (resp. ) for (resp. ). On the other hand is unirational, and we show that its coarse moduli is unirational and fibred over a rational base by homogeneous varieties which are rational if . Our study is motivated by the work of Gorchinskiy and Viviani on the moduli of hyperelliptic curves.

Paper Structure

This paper contains 15 sections, 33 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Stacks of simple cyclic covers of
  3. Rigidification of the stack of cyclic covers over a projective space
  4. Gerbe structure of over
  5. Degree of line bundles on the universal Brauer--Severi scheme
  6. Tautological family on Zariski open sets of the coarse moduli space of
  7. Existence of tautological families
  8. Picard group of $M_{n,r,d}^0$ and non existence of a tautological family over $M_{n,r,d}^0$
  9. Picard group of coarse moduli of the stack of cyclic covers over the projective space
  10. Stacks of cyclic triple covers over
  11. Rigidification of stacks of cyclic triple covers over
  12. Gerbe Structure of over
  13. Degree of line bundles on the universal Brauer--Severi
  14. Tautological families on Zariski open sets of the coarse moduli of
  15. Picard group of $M_{1,3,d_1,d_2}^0$ and non-existence of a tautological family on $M_{1,3,d_1,d_2}^0$

Key Result

Theorem 1.2

($=$ Theorem tautological family, Proposition tautological family in any dimension, Theorem Picard Group of the coarse moduli,) Let $M_{n,r,d}^0$ be the open subscheme of $M_{n,r,d}$ parametrizing simple cyclic covers of $\mathbb{P}^n$, whose automorphism group is $\mu_r$. Then the following stateme

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 34 more