We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The l-dimensional variety Ynl, which is the quotient of the product of a certain curve Cn
Nearby in the stack
by itself
l
times by a group
G≃(Z/nZ)l−1
of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If
n=3c
Schreieder constructed an explicit smooth birational model of it, and Flapan proved: the Kodaira dimension of this smooth model is 1, if
c>1
; if
l=2
it is a modular elliptic surface; if
l=3
it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on
n
and
l
we prove that
Ynl
admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if
l=2
its minimal resolution is a modular elliptic surface, obtained by a base change of order
n
on a specific extremal rational elliptic surface; if
l≥3
it has a birational model which admits a fibration in K3 surfaces and a fibration in
(l−1)
-dimensional varieties of Kodaira dimension at most 0.