Deformation of pairs and Noether–Lefschetz loci in toric varieties

Abstract

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 simplicial projective toric variety $$\mathbb {P}_{\Sigma }^{2k+1}$$ P Σ 2 k + 1 , with $$V\subset X$$ V ⊂ X . The hypersurface X is supposed to be of Macaulay type , which means that its toric Jacobian ideal is Cox–Gorenstein, a property that generalizes the notion of Gorenstein ideal in the standard polynomial ring. Under some assumptions, we prove that the class $$\lambda _V\in H^{k,k}(X)$$ λ V ∈ H k , k ( X ) deforms to an algebraic class if and only if it remains of type ( k , k ). 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 (Bruzzo and Montoya 15(2):682–694, 2021) and the main theorem proved by Dan (in: Analytic and Algebraic Geometry. Hindustan Book Agency, New Delhi, pp 107–115, 2017).