Fano 3-folds with divisible anticanonical class

Abstract

We show the nonvanishing of H0(X,−KX) for any a Fano 3-fold X for which −KX is a multiple of another Weil divisor in Cl(X). The main case we study is Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X)=1, \({\mathbb{Q}}\) -factorial terminal singularities and −KX = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties (X,A) and deduce both the nonvanishing of H0(X,−KX) and the sharp bound (−KX)3≥ 8/165. We find the families that can be realised in codimension up to 4.