manuscripta math. 154, 13–22 (2017) © Springer-Verlag Berlin Heidelberg 2016
· D. Loughran
The moduli of smooth hypersurfaces with level
Received: 14 January 2016 / Accepted: 1 December 2016
Published online: 19 December 2016
Abstract. We construct the moduli space of smooth hypersurfaces with level N structure
over Z[1/N ]. As an application we show that, for N large enough, the stack of smooth
hypersurfaces over Z[1/N ] is uniformisable by a smooth afﬁne scheme. To prove our results,
we use the Lefschetz trace formula to show that automorphisms of smooth hypersurfaces
act faithfully on their cohomology. We also prove a global Torelli theorem for smooth cubic
threefolds over ﬁelds of odd characteristic.
The moduli of smooth proper curves of genus g with g ≥ 2, or principally polarized
abelian schemes of ﬁxed dimension, or polarized K3 surfaces of ﬁxed degree are
smooth ﬁnite type separated Deligne–Mumford stacks over Z. All these stacks
admit level structures [23,30,31]. Such structures are usually introduced to help
rigidify the moduli problem and lead to interesting theory and applications .
The aim of this note is to construct a moduli stack of smooth hypersurfaces with
level structure. We will deﬁne a level N structure on a smooth hypersurface to be
a trivialization of its cohomology with Z/N Z-coefﬁcients (see Sect. 3 for details).
Key to our construction is the following result on the action of an automorphism
of a smooth hypersurface on its cohomology.
Theorem 1.1. Let d ≥ 3 and n ≥ 1 be integers with (d, n) = (3, 1). Let k be a ﬁeld
and let be a prime number which is invertible in k. Let X be a smooth hypersurface
of degree d in P
, and let σ ∈ Aut(X) be non-trivial. If char(k) = 0 or the order
of σ is coprime to char(k), then σ acts non-trivially on H
The question of whether the automorphism group of a variety acts faithfully
on its cohomology has been investigated for other families of varieties, such as
Enriques surfaces , hyperkähler varieties [4, Prop. 9], [9,§3],[30, §2.4], and
some surfaces of general type [11,28].
A. Javanpeykar (
): Institut für Mathematik, Johannes Gutenberg-Universität Mainz,
Staudingerweg 9, 55099 Mainz, Germany. e-mail: firstname.lastname@example.org
D. Loughran: School of Mathematics, University of Manchester, Oxford Road, Manchester
M13 9PL, UK.
Mathematics Subject Classiﬁcation: 14D23 (14K30, 14J50, 14C34)