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 P(n) 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 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). (C) 2010 Elsevier Inc. All rights reserved.