| Summary: | We construct families of polarised projective varieties as quasilinear sections of weighted flag varieties in codimensions 6,7,8,9 and 10. Our tools include a general formula for the Hilbert series of weighted flag varieties, an algorithm to compute their defining equations under their Plücker-style embeddings, and detailed computer-assisted searches. We study complete intersections in a weighted flag variety of the Lie group of type G2, the weighted Grassmannian wGr(2,6), the weighted Lagrangian Grassman- nian wLGr(3,6), a partial weighted flag variety of type A3 and finally a weighted flag variety of the exceptional Lie group of type F4. We construct many families of canon- ical Calabi-Yau 3-folds, log-terminal Q-Fano 3-folds and canonical 3-folds with cyclic quotient singularities.
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