Using the Grothendieck-Lefschetz theory ([SGA2]) and a generalization (due to Cutkosky) of a result of Grothendieck 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. 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 homogenous equations which define these scrolls set-theoretically.

This is joint work with G. Valla.