Codimension bounds for the Noether–Lefschetz components for toric varieties

Abstract

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