Deformation of pairs and Noether-Lefschetz loci in toric varieties

TL;DR

This paper investigates how the algebraic nature of subvarieties within hypersurfaces in toric varieties persists under deformation, linking cohomological classes to algebraic properties and extending previous results.

Contribution

It generalizes existing theorems by analyzing the deformation of pairs (V,X) in toric varieties and characterizing the Noether-Lefschetz locus as an irreducible component of a Hilbert scheme.

Findings

Cohomological classes remain of type (k,k) iff V stays algebraic during deformation.

The Noether-Lefschetz locus is an irreducible component of a Hilbert scheme.

Extends previous theorems to odd-dimensional simplicial projective toric varieties.

Abstract

We continue our study of the Noether-Lefschetz loci in toric varieties and investigate deformation of pairs (V,X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a odd dimensional simplicial projective toric variety, with V\subset X. Under some assumptions, we prove that the cohomological class in H^{k,k}(X) associated to V remains of type (k,k) under an infinitesimal deformation if and only if V remains algebraic. Actually we prove that locally the Noether-Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 in our previous work [4] and the main theorem proved by Dan in [10].