VARIETIES OF PICARD RANK ONE AS COMPONENTS OF AMPLE DIVISORS

Let V be an integral normal complex projective variety of dimension n >= 3 and denote by L an ample line bundle on V. By imposing that the linear system vertical bar L vertical bar contains an element A = A(1) + . . . + A(r), r >= 1, where all the Ai's are distinct effective Cartier divisors with Pic(A(i)) = Z, we show that such a V is as special as the components A(i) of A is an element of vertical bar L vertical bar. After making a list of some consequences about the positivity of the components Ai, we characterize pairs ( V, L) as above when either A(1) congruent to Pn-1 and Pic(A(j)) = Z for j = 2, . . ., r, or V is smooth and each A(i) is a variety of small degree with respect to [H-i] A(i), where [H-i] A(i) is the restriction to A(i) of a suitable line bundle H-i on V.