Codimension bounds for the Noether-Lefschetz components for toric varieties

For a quasi-smooth hyper-surface $X$ in a projective simplicial toric variety $P$, the morphism $i:H^p(P) \to H^p(X)$ induced by the inclusion is injective for $p=d$ and an isomorphism for $p<d-1$, where $d=dim\ P$. This allows one to define the Noether-Lefschetz locus $NL_{\beta}$ as the locus of quasi-smooth hypersurfaces of degree $\beta$ such that $i$ acting on the middle algebraic cohomology is not an isomorphism. In this paper we prove that, under some assumptions, if $dim P =2k+1$ and $k\beta-\beta_0=n\eta$ $(n\in\mathbb N)$, where $\eta$ is the class of a 0-regular ample divisor, and $\beta_0$ is the anticanonical class, then every irreducible component $V$ of the Noether-Lefschetz locus quasi-smooth hypersurfaces of degree $\beta$ satifies the bounds $n+1\leq codim\ V \leq h^{k-1,k+1}(X)$.


Introduction
The classical Noether-Lefschetz theory is about the Picard number of surfaces in 3-dimensional projective space. Let U d ⊂ PH 0 (P 3 , O P 3 (d)) be the locus of smooth surfaces of degree d in P 3 , with d ≥ 4; then the very general surface in U d has Picard number 1 (for an historical perspective of the Noether-Lefschetz problem, and exhaustive references the reader may consult [2]) . Moreover, if Z is a component of the locus in U d whose points correspond to surfaces with Picard number greater than 1 (the Noether-Lefschetz locus), then This result was generalized in [4,10] to quasi-smooth surfaces 1 in projective simplicial toric threefolds satisfying some conditions. The purpose of the present paper is to extend these bounds to the case of projective simplicial toric varieties of higher odd dimension, see Theorems 2.1 and 3.1 (when the ambient variety has even dimension the problem is trivial as the middle cohomology of hypersurfaces is controlled by the Lefschetz hyperplane theorem).
This short paper is a natural sequel to [6], where the definition of the Noether-Lefschetz locus was extended to simplicial projective toric varieties P 2k+1 Σ of arbitrary odd dimension. Given an ample class β in Pic(P 2k+1 Σ ), one considers sections f ∈ P(H 0 (O P 2k+1 Σ (β))) such that ) be the open subset parameterizing quasi-smooth hypersurfaces and let π ∶ χ β → U β be the tautological family. One considers the local system H 2k = R 2k π ⋆ C ⊗ O U β over U β . The associated flat connection (the Gauss-Manin connection) will be denoted by ∇.
Let 0 ≠ λ f ∈ H k,k (X f , Q) i * (H k,k (P 2k+1 Σ )) and let U be a contractible open subset around f . Finally, let λ ∈ H 2k (U ) be the section defined by λ f and letλ its image in In this paper we continue the study of the Noether-Lefschetz locus and establish lower and upper bounds for the codimension of its components. In section 2 we obtain the lower bound, which, following the terminology in [2], we call the "explicit Noether-Lefschetz theorem for toric varieties." In section 3, using the Hodge theory for hypersurfaces in complete simplicial toric varieties, and the orbifold structure of the quasi-smooth hyper-surfaces (see [1]), we establish the upper bound, extending the ideas in [4].
This research was partly supported by PRIN "Geometria delle varietà algebriche" and by GNSAGA-INdAM.

Explicit Noether-Lefschetz theorem in toric varieties
This section is a natural extension to higher dimensions of the ideas developed in [4,10] for the case of threefolds. To this end there are two points to consider: 1. Let S = ⊕ β S β be the Cox ring of the toric variety P 3 Σ under consideration. In [4,10] the following assumption was made. Let β and η be ample classes in Pic(P 3 Σ ), with η primitive and 0-regular (in the sense of Castelnuovo regularity), and β − β 0 = nη for some n ≥ 0, where β 0 is the anticanonical class of P 3 Σ . Then one assumes that the multiplication map S β ⊗ S nη → S β+nη is surjective; this implies that a very general quasi-smooth surface of degree β in P 3 Σ has the same Picard number as P 3 Σ . In the higher dimensional case, if we assume again the surjectivity of the multiplication map, using Theorem 10.13 and Proposition 13.7 in [1], and Lemma 3.7 in [3], one proves that the primitive cohomology of degree 2k of a very general quasi-smooth hypersurface of degree β is zero. Of course we recover the result of [8] when k = 1. [3,10] it alsowas assumed that

In
one to conclude that a certain vector bundle is 1-regular with respect to η. We assume and will prove the same regularity for that vector bundle.
The next Theorem establishes the lower bound for the codimension of the components of the Noether-Lefschetz locus.
Proof. The proof is a higher dimensional generalization of that in [4] (which in turn largely mimics the proof of [7,8] for the case of P 3 ), with the modification proposed in [10]. We take a base point free linear system W in H 0 (O P 2k+1 Σ (β)) and a complete flag of linear subspaces , which is locally free.
We have to prove that M 0 is 1-regular with respect to η, i.e., that H q (M 0 ((1 − q)η)) = 0 for every positive q (this is the regularity property we hinted at in the introduction). Taking cohomology from as π is surjective, H 1 (M 0 ) = 0. The vanishing of H q (M 0 (1 − q)η) = 0 for 1 < q ≤ 2k + 1 is obtained by induction, tensoring the short exact sequence (1) by O P 2k+1 Σ ((1 − q)η), and considering the segment of the long exact sequence of cohomology The rest of the proof follows as in [4,10].

Upper bound for the Codimension of the Noether-Lefschetz Components in Toric Varieties
The Explicit Noether-Lefschetz Theorem has provided a lower bound for the codimension of the Noether-Lefschetz components. Hodge theory in toric varieties will give us the upper bound. For a class β as in the previous Section, let f be a point in the Noether-Lefschetz locus, let X f be the corresponding hypersurface in P 2k+1 Σ , and let λ be a class as in Definition 1.1.
This section is devoted to proving this theorem. Classically it is a consequence of Griffiths' Transversality, which we want to extend to the context of projective simplicial toric varieties.
Variations of Hodge Structure. The tautological family π ∶ X β ⊂ U β × P Σ → U β is of finite type and separated since X β and U β are varieties. By Corollary 5.1 in [14] there exists a Zariski open set U ⊂ U β such that X = π −1 (U) → U is a locally trivial fibration in the classical topology, i.e., there exists an open cover of U by contractible open sets such that for every element U of the cover and every point X 0 ∈ U we have X U ≃ π −1 (U ) ≃ U × X 0 , which implies that X u ≃ X 0 for all u ∈ U as C ∞ orbifolds; moreover, H k (X u ) ≃ H k (X 0 ). Thanks to the locally trivialization and as quasi-smooth hypersurfaces are orbifolds [1], we can put an orbifold structure on X = π −1 (U ).
The Cartan-Lie formula. For every k, let H k be the complex vector bundle on U β associated to the local system R k π * C. Let Ω be a Zariski k-form on the orbifold X such that Ω u = Ω Xu is closed for every u ∈ U ; we can associate with it a local section ω of the vector bundle H k by letting The following result computes the Gauss-Manin connection ∇ ∶ H k → H k ⊗ Ω U in the direction w restricted to X 0 .
Proof. See [12]; actually the proof goes as in the classical case, see Proposition 9.2.2 in [16].
Again we take U a contractible open set trivializing X U U ≃ U × X 0 . is the map which to u ∈ U associates the term F p H k (X u , C) in the Hodge filtration of Here b p,k = dim F p H k (X u , C). Note that P p,k is a map of complex manifolds.
Proposition 3.4. The period map P p,k is holomorphic.
Proof. For the reader's convenience we sketch here a proof of this result, although it has been actually already proved in [12]. By Theorem 7.9 in [9] and the fact that Hodge theorem holds also in the orbifold case ( [13,18] and also section 2.1 in [11]) P p,k is a C ∞ map. The rest of the proof follows as in Theorem 10.9 in [16], whose strategy is to prove that the C-linear extension of the differential to T u U ⊗ C of P p,k vanishes on the vectors of type (0,1).
Remark 3.5. There is an intrinsic relation between the differential and the covariant derivative ∇ w ∶ H k → H k , namely, given σ ∈ F p H k (X u ) one can construct a local section of H k over Uσ such thatσ(u) = σ. Hence, dP p,k u (w)(σ) = ∇ wσ mod F p H k (X u ) △ Proposition 3.6 (Griffiths Transversality).
Proof. By the Cartan-Lie formula and the above remark The fact that P p,k is holomorphic implies that that ι v dΩ X 0 ∈ F p H k (X u ) if v is of type (0, 1), so that if v is of type (1, 0) we get ι v dΩ X 0 ∈ F p−1 H k (X u ).
Proof. Once Griffiths Transversality has been generalized, the proof goes as in classical case, see Lemma 3.1 in [15] and section 5.3 in [17]. This proves Theorem 3.1.