Existence and Density of General Components of the Noether–Lefschetz Locus on Normal Threefolds

Bruzzo, Ugo; Grassi, Antonella; Lopez, Angelo Felice

doi:10.1093/imrn/rnz358

Cited by 5 publications

(3 citation statements)

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“…For the projective space P 3 and more generally for projective normal 3-folds, the study of the components of the Noether-Lefschetz locus with maximal codimension has been studied by many authors [19,14,9,25]. The components of the Noether-Lefschetz locus with maximal codimension are called general components since they are dense in the Classical and in the Zariski topology.…”

Section: Components Of the Noether-lefschetz Locus With Maximal Codim...mentioning

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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Section: Components Of the Noether-lefschetz Locus With Maximal Codim...mentioning

confidence: 99%

Some open problems in Noether-Lefschetz theory for toric varieties

Montoya

2024

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“…it is enough to replace "dimension" by "period dimension" and the result is still true.) Regarding special cases the existence of components of NL Y of maximal codimension, we refer also to [5] and [3].…”

Section: Zilber-pink and The Noether-lefschetz Locus For Arbitrary Th...mentioning

confidence: 99%

On the distribution of the Hodge locus

Baldi,

Klingler,

Ullmo

2023

Invent. math.

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Given a polarizable ℤ-variation of Hodge structures $\mathbb{V}$ V over a complex smooth quasi-projective base $S$ S , a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of $S$ S , called the special subvarieties for $\mathbb{V}$ V . Our main result in this paper is that, if the level of ${\mathbb{V}}$ V is at least 3, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension (Theorem 1.5). For instance the Hodge locus of positive period dimension of the universal family of degree $d$ d smooth hypersurfaces in $\mathbf{P}^{n+1}_{\mathbb{C}}$ P C n + 1 , $n\geq 3$ n ≥ 3 , $d\geq 5$ d ≥ 5 and $(n,d)\neq (4,5)$ ( n , d ) ≠ ( 4 , 5 ) , is algebraic. On the other hand we prove that in level 1 or 2, the Hodge locus is analytically dense in $S^{\mbox{an}}$ S an as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in $S$ S of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension.

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“…One way to verify the hypothesis of Green's criterion is to construct Noether-Lefschetz loci of the expected dimension. Following Ciliberto and Lopez [14], we do so in Section 2 by considering Noether-Lefschetz loci associated to determinantal curves, a strategy independently adopted by Bruzzo, Grassi and Lopez in [11]. Finally, Section 3 contains the proof of Theorem 0.1.…”

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confidence: 99%

Sums of three squares and Noether–Lefschetz loci

Benoist¹

2018

Compositio Math.

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We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in P 3 whose function field has level 2 is dense in the set of those that have no real points. Strategy of the proofOur starting point is Colliot-Thélène's Hodge-theoretic proof of the Cassels-Ellison-Pfister theorem [15]: he associates to a polynomial f its homogenization F ∈ R[X 0 , X 1 , X 2 ], and the real algebraic surface defined by S := {Z 2 + F (X 0 , X 1 , X 2 ) = 0}. He then interpretes the polynomials f that are sums of three squares in R(x 1 , x 2 ) as those for which the complex surface S C carries an extra line bundle of a particular kind, and concludes by applying the Noether-Lefschetz theorem.As a consequence, Q d may be viewed as a union of Noether-Lefschetz loci in P d . Over C, density results for Noether-Lefschetz loci have been first obtained by Ciliberto-Harris-Miranda and Green [13], and we adapt these arguments over R.We rely on a real analogue of Green's infinitesimal criterion ([13, §5], [37, §17.3.4]), which was developed for other purposes in a joint work with Olivier Wittenberg [4]. Section 1 is devoted to establishing this criterion in a form suitable for our needs: Proposition 1.3. One way to verify the hypothesis of Green's criterion is to construct Noether-Lefschetz loci of the expected dimension. Following Ciliberto and Lopez [14], we do so in Section 2 by considering Noether-Lefschetz loci associated to determinantal curves, a strategy independently adopted by Bruzzo, Grassi and Lopez in [11]. Finally, Section 3 contains the proof of Theorem 0.1. Level of function fieldsThe argument described above may be adapted to other families of real surfaces: here is another application of it. Recall that if K is a field, Pfister [32, Satz 4] has shown that the smallest integer s such that −1 is a sum of s squares in K is a power of 2 (or +∞): it is the level s(K) of K. Moreover, if X is an integral variety over R of dimension n without real points, s(R(X)) 2 n [33, Theorem 2].Let us restrict to varieties that are smooth degree d surfaces S ⊂ P 3 R , defined by a degree d homogeneous equation F ∈ R[X 0 , X 1 , X 2 , X 3 ] d . Let Θ d ⊂ P(R[X 0 , X 1 , X 2 , X 3 ] d ) be the set of those surfaces that have no real points. As before, we assume that d is even, since otherwise Θ d = ∅. If d = 2, any such surface S is isomorphic to the anisotropic quadric {X 2 0 +X 2 1 +X 2 2 +X 2 3 = 0}, and s(R(S)) = 2. On the other hand, it follows from the Noether-Lefschetz theorem applied as in [15] that if d 4, a very general S ∈ Θ d satisfies s(R(S)) = 4. In section 4, we will show: Conventions about algebraic varieties over RAn algebraic variety X over R is a separated scheme of finite type over R. We denote its complexification by X C . Its set of complex points X(C) is endowed with an action of G := Gal(C/ R) ≃ Z /2 Z such that the complex conjugation σ ∈ G acts antiholomorphically. The r...

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Existence and Density of General Components of the Noether–Lefschetz Locus on Normal Threefolds

Cited by 5 publications

References 35 publications

Some open problems in Noether-Lefschetz theory for toric varieties

Some open problems in Noether-Lefschetz theory for toric varieties

On the distribution of the Hodge locus

Sums of three squares and Noether–Lefschetz loci

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