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

Abstract

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 P n 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).