Stacks of cyclic covers of projective spaces

Abstract

We define stacks of uniform cyclic covers of Brauer-Severi schemes, proving that they can be realized as quotient stacks of open subsets of representations, and compute the Picard group for the open substacks parametrizing smooth uniform cyclic covers. Moreover, we give an analogous description for stacks parametrizing triple cyclic covers of Brauer-Severi schemes of rank 1 that are not necessarily uniform, and give a presentation of the Picard group of the substacks corresponding to smooth triple cyclic covers.

The Picard group of the moduli stack of stable hyperelliptic curves

“…Moreover, it was proved in the same paper that the boundary classes are independent in Pic(H g ) ⊗ Q. We wish to show that, combining these results of [2] with those of [1] and an idea of [3], one gets almost immediately a complete description of Pic(H g ); in particular, one finds that (1) is valid already in Pic(H g ), and not just modulo torsion. One also gets an alternate-and to me somewhat simpler-proof of the result of Gorchinskiy and Viviani mentioned above.…”

“…On the other hand, since hλ restricts to the class of a trivial line bundle on H g , it must be an integral linear combination of boundary classes. Since the boundary classes are independent, this relation must be proportional to (1). Thus the integer (8g + 4)/ h divides both g and 8g + 4 = 4(2g + 1).…”

“…In a recent paper [1], Arsie and Vistoli have shown, as a byproduct of their study of moduli of cyclic covers of projective spaces, that the Picard group of the moduli stack H g of smooth hyperelliptic curves of genus g ≥ 2 is finite cyclic, and that its order is 8g + 4 for odd g, and 4g + 2 for even g, over any field of characteristic not dividing 2g + 2. In another recent paper [3], Gorchinskiy and Viviani have given a geometric construction of generators for the Picard groups in question.…”

A note on families of hyperelliptic curves

“…The stacks H g and D 2g+2 admit the following description as quotient stacks (see [AV04,Cor. 4.7] and [GV08, Prop.…”

“…In Theorem 3.1 we give a different answer to these questions in terms of the geometry of the base S. Our proof is completely stack-theoretic and uses the fact that the stack H g of hyperelliptic curves is a µ 2 -gerbe over the stack D 2g+2 of conic bundles endowed with an effective Cartier divisor finite andétale of degree 2g + 2, and the fact that both these stacks have an explicit description as quotient stacks (see [AV04] and [GV08]). …”