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We give the construction of a codimension 10 weighted homogeneous variety$w\Sigma F_4(\mu,u)$ corresponding to the exceptional Lie group $F_4$ byexplicit computation of its graded ring structure. We give a formula for theHilbert series of the generic weighted $w\Sigma F_4(\mu,u)$ in terms ofrepresentation theoretic data of $F_4$. We also construct some families ofpolarized 3-folds in codimension 10 whose general member is the weightedcomplete intersection of some $w\Sigma F_4(\mu,u)$.

Polarized 3-folds in a codimension 10 weighted homogeneous $F_4$ variety