The family of lines on the Fano threefold <i>V</i>₅

Abstract

A smooth projective algebraic 3-fold V over the field C is called a Fano 3-fold if the anticanonical divisor — K v is ample. The integer g = g(V) = ½(- K v ) 3 is called the genus of the Fano 3-fold V . The maximal integer r ≧ 1 such that ϑ(— K v )≃ ℋ r for some (ample) invertible sheaf ℋ ε Pic V is called the index of the Fano 3-fold V . Let V be a Fano 3-fold of the index r = 2 and the genus g = 21 which has the second Betti number b 2 (V) = 1. Then V can be embedded in P 6 with degree 5, by the linear system |ℋ| , where ϑ(— K v ) ≃ ℋ 2 (see Iskovskih [5]). We denote this Fano 3-fold V by V 5 .