Some remarks on morphisms between Fano threefolds

Some twenty-five years ago, Iskovskih classified the smooth complex Fano threefolds with Picard number one. Apart from P and the quadric, his list includes 5 families of Fano varieties of index two and 11 families of varieties of index one (for index one threefolds, the cube of the anticanonical divisor takes all even values from 2 to 22, except 20). Recently, the author ([A]) and C. Schuhmann ([S]) made some efforts to classify the morphisms between such Fano threefolds, the starting point being a question of Peternell: let f : X → Y be a non-trivial morphism between Fano varieties with Picard number one, is it then true that the index of X does not exceed the index of Y ? In particular, Schuhmann ([S]) proved that there are no morphisms from indextwo to index-one threefolds, and that any morphism between index-two threefolds is an isomorphism (under certain mild additional hypotheses, some of which were handled later in [A], [IS]). As for morphisms from index-one to index-two Fano threefolds, such morphisms do exist: an index-two threefold has a double covering (branched along an anticanonical divisor) which is of index one. It is therefore natural to ask if every morphism from index-one Fano threefold X with Picard number one to index-two Fano threefold Y with Picard number one is a double covering. In [A], I proved a theorem (Theorem 3.1) indicating that the answer should be yes, however not settling the question completely. The essential problem was that the methods of [A] would never work for Y = V5, the linear section of the Grassmannian G(1, 4) in the Plücker embedding (all smooth three-dimensional linear sections of G(1, 4) are isomorphic). Though there are several ways to obtain bounds for the degree of a morphism between Fano threefolds with second Betti number one ([HM], [A]), these bounds are still too rough for our purpose. This paper is an attempt to handle this problem. The main result is the following