An asymptotic description of the NoetherLefschetz components in toric varieties
Abstract
We extend the definition of NoetherLeschetz components to quasismooth hypersurfaces in a projective simplicial toric variety $\mathbb P_{\Sigma}^{2k+1}$, and prove that asymptoticaly the components whose codimension is bounded from above by a suitable effective constant correspond to hypersurfaces containing a small degree $k$dimensional subvariety. As a corollary we get an asymptotic characterization of the components with small codimension, generalizing the work of Otwinowska for $\mathbb P_{\Sigma}^{2k+1}=\mathbb P^{2k+1}$ and Green and Voisin for $\mathbb P_{\Sigma}^{2k+1}=\mathbb P^3$. Some tools that are developed in the paper are a generalization of Macaulay's theorem for Fano, irreducible normal varieties with rational singularities, satisfying a suitable additional condition, and an extension of the notion of Gorenstein ideal for normal varieties with finitely generated Cox ring.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.01570
 Bibcode:
 2019arXiv190501570B
 Keywords:

 Mathematics  Algebraic Geometry;
 14C22;
 14J70;
 14M25
 EPrint:
 22 pages. v2: Some definitions clarified