Deformation of pairs and Noether–Lefschetz loci in toric varieties

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 PΣ2k+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}_{\Sigma }^{2k+1}$$\end{document}, with V⊂X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\subset X$$\end{document}. 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 λV∈Hk,k(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _V\in H^{k,k}(X)$$\end{document} 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).


Introduction
In this short note we continue our study of the Noether-Lefschetz loci in toric varieties and investigate the deformation of pairs (V, X) where V is a k-dimensional complete intersection subvariety and X a quasi-smooth ample hypersurface in a simplicial projective toric variety P 2k+1 Σ of odd dimension 2k + 1 ≥ 3, with V ⊂ X.We assume that the local Noether-Lefschetz locus N L k,β λ V ,U , also called "Hodge locus" in the literature when P 2k+1 Σ is a projective space, as defined in Section 5, is not empty (a condition for this to happen is for instance given in Lemma 3.7 of [2]).Here λ V is the cohomology class of V , and β is the class of X in Pic(P 2k+1 Σ ).Then every irreducible component L of the full Noether-Lefschetz locus N L β , namely the locus in the linear system β of the points corresponding to quasi-smooth hypersurfaces whose (k, k)cohomology does not come entirely from the ambient P 2k+1 Σ , is locally the Hodge locus [3].In other terms, there exists an open subset U of β such that L ∩ U = N L k,β λ V ,U .Moreover, under the further assumption that β satisfies β = q η + β ′ (n ∈ N), where q ∈ Q >0 , η is a primitive ample class in P 2k+1 Σ , and β ′ is a nef Cartier class, if X cointains a k-dimensional complete intersection subvariety with deg η V < q, we will show that under infinitesimal deformations, V is algebraic if and only if its associated cohomology class λ This extends the work of Dan in [10] and the last result of [4] (Theorem 4.2) for toric varieties with higher Picard rank (there the Picard number was assumed to be one, and moreover, the result is asymptotic).Definition 2.2.An infinitesimal variation of Hodge structure {H Z , Hp, q, Q, T, δ} is given by a polarized Hodge structure together with a vector space T and linear map δ ∶ T → ⊕ 1≤p≤n Hom(H p,q , H p−1,q+1 ) that satisfies the two conditions: Here F • is the filtration of H n given by For a quasi-smooth hypersurface X in a simplicial projective toric variety, δ is the morphism associated via tensor-hom adjuction to γ = ∑ p γ p , where is the natural multiplication map; for more details see 3.3 in [2].Given an infinitesimal variation of Hodge structure of weight 2k, there is an invariant associated to Let us assume γ is the primitive part of the class of k-codimensional algebraic cycle One has the following fact ( [11], Observation 4.a.4).
This is the result that we shall need later on.

A dual Macaulay theorem in toric varieties
We review a theorem of Macaulay theorem proved in [4].We refer the reader to that paper for all proofs of results in this section.
The Cox ring S of a complete simplicial toric variety P Σ is graded over the effective classes in the class group Cl(P Σ ) (see e.g.[8]).As O P Σ (α) is coherent and P Σ is complete, each S α is finite-dimensional over C; in particular, S 0 ≃ C. For every effective N ∈ Cl(P Σ ), the set of classes α ∈ Cl(P Σ ) such that N − α is effective is finite.
We shall give a definition of Cox-Gorenstein ideal of the Cox rings which generalizes to toric varieties the definition given by Otwinowska in [12] for projective spaces.Let B ⊂ S be the irrelevant ideal, and for a graded ideal I ⊂ B, denote by V T (I) the corresponding closed subscheme of P Σ .Definition 3.1.A graded ideal I of S contained in B is said to be a Cox-Gorentstein ideal of socle degree N ∈ Cl(P Σ ) if 1. there exists a C-linear form Λ ∈ (S N ) ∨ such that for all α ∈ Cl(P Σ ) 2. the natural bilinear morphism (called "Poincaré duality") is nondegenerate whenever α and N − α are effective.
We shall need to use a Euler form on P Σ [1,9,7].We denote by M the dual lattice of the lattice N which contains the fan Σ, i.e., Σ ⊂ N ⊗ R. Let d = dim P Σ .For more details about these definitions see [1].
Let f 0 , . . ., f d be homogeneous polynomials with deg(f i ) = α i , where each α i is ample, and let N = ∑ i α i − β 0 ; here β 0 is the anticanonical class of P Σ .Assume that the f i have no common zeroes in P Σ , i.e., V T (f 0 , . . ., f d ) = ∅.For each G ∈ S N one can define a meromorphic d-form ξ G on P Σ by letting where Ω 0 is a Euler form on P Σ .The form ξ G determines a class in H d (P Σ , ω), where ω is the canonical sheaf of P Σ , i.e., the sheaf of Zariski d-forms on P Σ , and the trace morphism Tr P Σ ∶ H d (P Σ , ω) → C associates a complex number to G. We define Λ ∈ (S N ) ∨ as We can now state a toric version of Macaulay's theorem.
Theorem 3.5.The linear map defined in Eq. (1) satisfies the condition in Definition 3.1.Therefore, the ideal Examples of Cox-Gorenstein ideals may be given in terms of toric Jacobian ideals.For every ray ρ ∈ Σ(1) denote by v ρ its rational generator, and by x ρ the corresponding variable in the Cox ring.Recall that d is the dimension of the toric variety P Σ , while we denote by r = #Σ(1) the number of rays.Given f ∈ S β one defines its toric Jacobian ideal as J 0 (f ) = x ρ 1 ∂f ∂x ρ 1 , . . ., x ρr ∂f ∂x ρr .
We recall from [1] the definition of nondegenerate hypersurface and some properties (Def.4.13 and Prop.4.15).Definition 3.6.Let f ∈ S β , with β an ample Cartier class.The associated hypersurface X f is nondegenerate if for all σ ∈ Σ the affine hypersurface X f ∩ O(σ) is a smooth codimension one subvariety of the orbit O(σ) of the action of the torus T d .Proposition 3.7.
2. If f is generic then X f is nondegenerate.
1.The toric Jacobian ideal of f coincides with the ideal 2. The following conditions are equivalent: (a) f is nondegenerate; (b) the polynomials x ρ i ∂f ∂xρ i , i = 1, . . ., r, do not vanish simultaneously on X f ; (c) the polynomials f and 4 The tangent space to the Noether-Lefschetz locus From now we assume d = 2k + 1.Let N = (k + 1)β − β 0 and let J 0 (f ) be the toric Jacobian ideal associated to f , which is Cox-Gorenstein of socle degree 2N + β 0 .Then there is a perfect pairing and by T 0 its lift in S N , where P is a preimage of γ under the natural map Definition 4.1.T ⊂ S be the Cl(Σ)-graded module such that T α is the largest subspace where T α ⊗ S N −α is contained in T 0 for α ≤ N , T N = T 0 and E N +α = T 0 ⊗ S α for α ≥ 0.
Remark 4.2.Note that T is a Cox-Gorentein ideal with socle degree N .△ Actually T β is the tangent space of the local Noether-Lefschetz locus at f : Proof.A superimposed bar will denote the class in R = S J of an element in S. Now, using Poincaré duality that means x 0 ...x r P H = 0 in R N +β+β 0 and equivalently P H = 0 in R N +β if and only if H ∈ T f N L λ,β .(see Theorem 6.2 in [5]).
Let us suppose that V is the zero locus of < A 1 , . . ., A k+1 > and since V ⊂ X f there exist Now we state and prove the main result of this paper.
Theorem 5.3.Assume that β is as in the Lemma.Let V be a quasi-smooth intersection in P 2k+1 Σ of codimension k + 1 and let X be a quasi-smooth hyper-surface of class β containing V such that deg η V < q.Assume also that the Noether-Lefschetz locus N L λ V β is nonempty.Then, λ V deforms to a (k, k) class if and only if λ [V ] deforms to an algebraic cycle.
In particular, N L k,β λ V ,U is isomorphic to an irreducible component of pr 2 (Hilb P,Q ), where P , Q are the Hilbert polynomials of V and X, respectively.
Proof.By the assumption on the degree of V , one has pr 2 (Hilb P,Q ) ⊂ N L k,β λ V ,U .Then, codim U pr 2 (Hilb P,Q ) ≥ codim U N L k,β λ V ,U ≥ codim T X U T X N L k,β λ V ,U .
On the other hand, keeping in mind that T β = I β ⊂ I β V , we have a natural map φ from T β to Hilb P,Q , which sends a homogeneous polynomial of degree β to its zero locus.One has Im(φ) ⊂ pr 2 (Hilb P,Q ) and since the zero locus is invariant under the torus action, dim T β > dim Im(φ).Hence, codim pr 2 (Hilb P,Q ) ≤ codim Im(φ) ≤ codim T β = codim T X N L k,β λ V ,U .
So pr 2 (Hilb P,Q ) and N L k,β λ V ,U have the same dimension, which implies the claim.

Proof.
We shall denote by (Z) the class in A d (P 2k+1 Σ ) of a d-dimensional closed subvariety Z of Pic(P 2k+1 Σ ), and by [Z] its class in A 2k+1−d (P 2k+1 Σ ).By hypothesis we have V = X ∩ W for a closed (k + 1)-dimensional subvariety W of P 2k+1 Σ .Thus we have