On the codimension of Noether–Lefschetz loci for toric threefolds

Lanza, Valeriano; Martino, Ivan

doi:10.1142/s0219199719500378

Cited by 2 publications

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“…This section is a natural extension to higher dimensions of the ideas developed in [7,22] for the case of threefolds. To this end there are two points to consider:…”

Section: An Explicit Noether-lefschetz Theorem In Toric Varietiesmentioning

confidence: 99%

Some open problems in Noether-Lefschetz theory for toric varieties

Montoya

2024

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On a quasi-smooth hypersurface X in an n-dimensional projective simplicial toric variety P n Σ associated to a fan Σ, the morphism i * : H p (P n Σ , Q) → H p (X, Q) induced by the inclusion, is injective for p = dim X and an isomorphism for p ≤ dim X −1. When n = 2k+1 one can define the Noether-Lefschetz locus N L β as the locus of quasi-smooth hypersurfaces of degree β such that i * restricted to the middle algebraic cohomology is not an isomorphism. In [6] Bruzzo and Grassi proved a Noether-Lefschetz type theorem: if the projective simplicial toric variety is Oda, i.e., the multiplication morphismis surjective whenever α and γ are an ample and nef class, respectively, then, on a very general hypersurface X,a countable union of closed subschemes in the projective complete linear system PH 0 (O P 2k+1 Σ (β)). The aims of this review article are to give a brief survey and to present some open problems of the Noether-Lefschetz locus and its components.

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“…This section is a natural extension to higher dimensions of the ideas developed in [7,22] for the case of threefolds. To this end there are two points to consider:…”

Section: An Explicit Noether-lefschetz Theorem In Toric Varietiesmentioning

confidence: 99%

Some open problems in Noether-Lefschetz theory for toric varieties

Montoya

2024

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show abstract

“…This section is a natural extension to higher dimensions of the ideas developed in [5,13] for the case of threefolds. To this end there are two points to consider:…”

Section: Explicit Noether-lefschetz Theorem In Toric Varietiesmentioning

confidence: 99%

“…be the Cox ring of the toric variety P 2k+1 under consideration. In [5,13] the following assumption was made in the case k = 1. Let β and η be ample classes in Pic(P 3 ), 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 .…”

Section: Explicit Noether-lefschetz Theorem In Toric Varietiesmentioning

confidence: 99%

“…Proof The proof is a higher dimensional generalization of that in [5] (which in turn largely mimics the proof of [9,10] for the case of P 3 ), with the modification proposed in [13]. We take a base point free linear system W in H 0 (P 2k+1 , O P 2k+1 (β)) and a complete flag of linear subspaces…”

Section: Explicit Noether-lefschetz Theorem In Toric Varietiesmentioning

confidence: 99%

“…This result was generalized in [5,13] to quasi-smooth surfaces in projective simplicial toric threefolds satisfying some conditions. 1 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).…”

Section: Introductionmentioning

confidence: 95%

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Codimension bounds for the Noether–Lefschetz components for toric varieties

Bruzzo

Montoya

2021

European Journal of Mathematics

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For a quasi-smooth hypersurface X in a projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$ P Σ , the morphism $$i^*:H^p(\mathbb {P}_{\Sigma })\rightarrow H^p(X)$$ i ∗ : H p ( P Σ ) → H p ( X ) induced by the inclusion is injective for $$p=\dim X$$ p = dim X and an isomorphism for $$p<\dim X-1$$ p < dim X - 1 . This allows one to define the Noether–Lefschetz locus $$\mathrm{NL}_{\beta }$$ NL β as the locus of quasi-smooth hypersurfaces of degree $$\beta $$ β such that $$i^*$$ i ∗ acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if $$\dim \mathbb {P}_{\Sigma }=2k+1$$ dim P Σ = 2 k + 1 and $$k\beta -\beta _0=n\eta $$ k β - β 0 = n η , $$n\in \mathbb {N}$$ n ∈ N , where $$\eta $$ η is the class of a 0-regular ample divisor, and $$\beta _0$$ β 0 is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree $$\beta $$ β satisfies the bounds $$n+1\leqslant \mathrm{codim}\,Z \leqslant h^{k-1,\,k+1}(X)$$ n + 1 ⩽ codim Z ⩽ h k - 1 , k + 1 ( X ) .

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On the codimension of Noether–Lefschetz loci for toric threefolds

Cited by 2 publications

References 7 publications

Some open problems in Noether-Lefschetz theory for toric varieties

Some open problems in Noether-Lefschetz theory for toric varieties

Codimension bounds for the Noether–Lefschetz components for toric varieties

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