Grothendieck-Lefschetz Theory, Set-Theoretic Complete Intersections and Rational Normal Scrolls

Using the Grothendieck-Lefschetz theory (see \cite{[SGA2]}) we prove a criterion to deduce that certain subvarieties of ${\mathbb{P}}^n$ of dimension $\ge 2$ 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 $S_{n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_d}$ 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).