  The aim of this paper is to present a construction of $t$-divisible designs
for $t>3$, because such divisible designs seem to be missing in the literature.
To this end, tools such as finite projective spaces and their algebraic
varieties are employed. More precisely, in a first step an abstract
construction, called $t$-lifting, is developed. It starts from a set $X$
containing a $t$-divisible design and a group $G$ acting on $X$. Then several
explicit examples are given, where $X$ is a subset of $PG(n,q)$ and $G$ is a
subgroup of $GL_{n+1}(q)$. In some cases $X$ is obtained from a cone with a
Veronesean or an $h$-sphere as its basis. In other examples $X$ arises from a
projective embedding of a Witt design. As a result, for any integer $t\geq 2$
infinitely many non-isomorphic $t$-divisible designs are found.
