Grothendieck–Lefschetz theory, set-theoretic complete intersections and rational normal scrolls

Using the Grothendieck–Lefschetz theory (see Grothendieck, 1968 [15], ) and a generalization (due to Cutkosky, 1997 [10], ) of a result from Grothendieck (1968) [15] concerning the simple connectedness, we prove that many closed subvarieties of Pn of dimension ⩾2 need at least n−1 equations to be defined in Pn set-theoretically, i.e. their arithmetic rank is ⩾n−1 (Theorem 1 of the Introduction). As applications we give a number of relevant examples. In the second part of the paper we prove that the arithmetic rank of a rational normal scroll of dimension d⩾2 in PN 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).