Noether-Lefschetz Theory in Toric Varieties

In 1994 Batyrev and Cox proved the "Lefschetz hyper-surface theorem" for toric varieties, which claims that for a quasi-smooth hyper-surface $X={f=0}$ in a complete simplicial toric variety $Pj^{2k+1}_{Sigma}$ the morphism $i^*:H^p(Pj_{Sigma}) ightarrow H^p(X)$ induced by the inclusion, is injective for $p=2k$ and an isomorphism for $p<2k$. This allows us to define $NL_{eta}$, the main geometrical object of this work, the locus of quasi-smooth hypersurfaces of degree $eta$ such that $i^*$ is not an isomorphism. Following the tradition we call it in cite{bm} the Noether-Lefschetz locus, while some authors call it Hodge loci when $Pj^{2k+1}_{Sigma}=Pj^{2k+1}$. This is a interesting geometrical object since it is the locus where the Hodge Conjecture is unknown. The cornerstone of this thesis, a Noether-Lefschetz theorem, is a consequence of "the infinitesimal Noether-Lefschetz theorem" namely, Bruzzo and Grassi in 2012 showed that if the multiplication $R(f)_{eta}otimes R(f)_{keta-eta_0} ightarrow R(f)_{(k+1)eta-eta_0} $ is surjective, where $eta_0$ is the class of the anticanonical divisor of $Pj^{2k+1}_{Sigma}$, the Noether-Lefschetz locus is non-empty and each irreducible component has positive codimension. We prove in Chapter 2 that if $keta-eta_0=neta$ $(ninN)$ where $eta$ is the class of an ample, primitive and $0$-regular divisor and $eta$ is $0$-regular with respect to $eta$, then every irreducible component $N$ of the Noether-Lefschetz locus respect to $eta$ satisfies $n+1leq codim N leq h^{k-1,k+1}(X)$. The lower bound generalize to higher dimensions some of the work of Green , Voisin and Lanza and Martino and the upper bound extend some results of Bruzzo and Grassi in 2018. In Chapter 3, continuing the study of the Noether-Lefschetz components, we prove that asymptotically the components whose codimension is bounded from above are made of hypersurfaces containing a small degree $k$-dimensional subvariety $V$. As a corollary we get an asymptotic characterization of the components of small codimension, generalizing the work of Otwinowska in 2003 for $Pj_{Sigma}^{2k+1}=Pj^{2k+1}$, Green and Voisin for $Pj_{Sigma}^{2k+1}=Pj^3$. Finally in chapter 4 we prove asymptotically the Hodge Conjecture when $V$ as before is smooth complete intersection. We also prove that on a very general quasi-smooth intersection subvariety in a projective simplicial toric variety the Hodge conjecture holds. We end this work with a natural and different extension of the Noether-Lefschetz loci. Some tools that have been developed in the thesis are a generalization of Macaulay theorem for Fano, irreducible normal varieties with rational singularities, satisfying a suitable additional condition, and an extension of the notion of Gorenstein ideal to toric varieties.