The moduli of smooth hypersurfaces with level structure (Q2408122)

The authors extend some of their earlier results [Math. Ann. 368, No. 3--4, 1191--1225 (2017; Zbl 1427.11062)] to positive characteristic and at the same time refine them so as to work over \(\mathrm{Spec }{\mathbb Z}[1/N]\) rather than after some base change. Suppose \(X\) is a smooth hypersurface of degree \(d\) in \({\mathbb P}^{n+1}\) over a field \(k\) in which the prime \(\ell\) is invertible. A tame automorphism is a non-trivial \(\sigma\in\mathrm{Aut }(X)\) whose order is prime to \(\mathrm{char }k\) if the characteristic is positive. A preliminary result here is that such a \(\sigma\) acts non-trivially on \(H^n_{\mathrm{\'et}}(X_{\bar k},{\mathbb Q}_\ell)\), apart from a few very obvious exceptions in low degree. The point is that every such automorphism extends from a linear map and in that case one can use the Lefschetz trace formula to compute the effect on cohomology. The control over stabilisers that this result gives, combined with standard facts about quotient stacks, allows the authors to define an algebraic stack over \({\mathbb Z}[1/N]\) parametrising hypersurfaces with level-\(N\) structure, and to show that this stack is in fact an affine scheme over \({\mathbb Z}[1/N]\). An application of this is a proof of a Torelli theorem for cubic threefolds in arbitrary characteristic different from 2, again extending their earlier results.