Resumen de Fano 3-folds with divisible anticanonical class

Gavin Brown , Kaori Suzuki

• We show the nonvanishing of H 0(X,-K X ) for any a Fano 3-fold X for which -K X 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, -factorial terminal singularities and -K X = 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 H 0(X,-K X ) and the sharp bound (-K X )3= 8/165. We find the families that can be realised in codimension up to 4.