of homology there exists a ( k + 2 ) -dimensional algebraic subvariety V ⊂ P 2 k + 1 Σ ,X such that V ∩ Y = C j so they are equal as a homology class of P 2 k + 1 Σ ,X ,i.e., [ V ∩ Y ] = [ C j ] . It is easy to check that π ( V ) ∩ X = C j as a subvariety of P k + 2 Σ where π ∶ ( x, y ) ↦ x . Hence [ π ( V ) ∩ X ] = [ C j ] which is equivalent to say that λ C j comes from P k + 2 Σ which contradicts the choice of [ C j ] .\n\nRemark 5.2 . Into the proof of the previous theorem, the key fact was that on X the Hodge conjecture holds and we translate it to Y by contradiction. So, using an analogous argument we have:\n\nargument we have:\n\nProposition 5.3. Let Y = { F = y 1 f s +⋯+ y s f s = 0 } ⊂ P 2 k + 1 Σ ,X be the quasi-smooth hypersurface associated to a quasi-smooth intersection subvariety X = X f 1 ∩ ⋅ ⋅ ⋅ ∩ X f s ⊂ P d Σ such that d + s = 2 ( k + 1 ) . If the Hodge conjecture holds on X then it holds as well on Y .\n\nCorollary 5.4. If the dimension of Y is 2 s − 1 , 2 s or 2 s + 1 then the Hodge conjecture holds on Y .\n\nProof. By Proposition 5.3 and
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