on X\n\nProof. If the dim C X = 1 the result is clear by the Hard Lefschetz theorem for projective orbifolds. The dimension 2 and 3 cases are covered by Theorem 3.5 and the Hard Lefschetz.\n\nCayley trick and Cayley proposition\n\nThe Cayley trick is a way to associate to a quasi-smooth intersection subvariety a quasi- smooth hypersurface. Let L 1 , . . . , L s be line bundles on P d Σ and let π ∶ P ( E ) → P d Σ be the projective space bundle associated to the vector bundle E = L 1 ⊕ ⋯ ⊕ L s . It is known that P ( E ) is a ( d + s − 1 ) -dimensional simplicial toric variety whose fan depends on the degrees of the line bundles and the fan Σ. Furthermore, if the Cox ring, without considering the grading, of P d Σ is C [ x 1 , . . . , x m ] then the Cox ring of P ( E ) is\n\nMoreover for X a quasi-smooth intersection subvariety cut oﬀ by f 1 , . . . , f s with deg ( f i ) = [ L i ] we relate the hypersurface Y cut oﬀ by F = y 1 f 1 + ⋅ ⋅ ⋅ + y s f s which turns out to be quasi-smooth. For more details see Section 2 in [7].\n\nWe will denote P ( E ) as P d + s − 1 Σ ,X to keep track of its relation with X and P d Σ .\n\nThe following is a key remark.\n\nRemark 4.1 . There is a morphism ι ∶ X → Y
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