Codimension bounds for the Noether–Lefschetz components for toric varieties

Abstract For a quasi-smooth hypersurface X in projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$ P Σ , the morphism $$i^*:H^p(\mathbb })\rightarrow H^p(X)$$ i ∗ : H p ( ) → X induced by inclusion is injective for $$p=\dim X$$ = dim and an isomorphism $$p<\dim X-1$$ < - 1 . This allows one to define Noether–Lefschetz locus $$\mathrm{NL}_{\beta NL β as of hypersurfaces degree $$\beta $$ such that $$i^*$$ acting on middle algebraic cohomology not isomorphism. We prove that, under some assumptions, if $$\dim \mathbb }=2k+1$$ 2 k + $$k\beta -\beta _0=n\eta 0 n η $$n\in {N}$$ ∈ N where $$\eta class 0-regular ample divisor, _0$$ anticanonical class, every irreducible component V satisfies bounds $$n+1\leqslant \mathrm{codim}\,Z \leqslant h^{k-1,\,k+1}(X)$$ ⩽ codim Z h ,