Polarized 3-folds in a codimension 10 weighted homogeneous F4 variety

We describe the construction of a codimension 10 weighted homogeneous variety wΣF4(μ,u) corresponding to the exceptional Lie group F4 by explicit computation of its graded ring structure. We give a formula for the Hilbert series of the generic weighted wΣF4(μ,u) in terms of representation theoretic data of F4. We also construct some families of polarized 3-folds in codimension 10 whose general member is a weighted complete intersection of some wΣF4(μ,u).