  Using the Grothendieck-Lefschetz theory (see \cite{[SGA2]}) we prove a
criterion to deduce that certain subvarieties of $\mathbb P^n$ of dimension
$\geq 2$ are not set-theoretic complete intersections (see Theorem 1 of the
Introduction). As applications we give a number of relevant examples. In the
last part of the paper we prove that the arithmetic rank of a rational normal
$d$-dimensional scroll $S_{n_1,...,n_d}$ in $\mathbb P^N$ is $N-2$, by
producing an explicit set of $N-2$ homogeneous equations which define these
scrolls set-theoretically (see Theorem 2 of the Introduction).
