Picard numbers of quintic surfaces

Schütt, Matthias

doi:10.1112/plms/pdu056

Cited by 7 publications

(4 citation statements)

References 33 publications

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“…Therefore, we can focus on the following subset of R g : 1≤n≤g−1 R n + R g−n . 16 Now we need a little bit of notation. Let us set R g,n := (g − n) 2…”

Section: 2mentioning

confidence: 99%

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On the Picard numbers of abelian varieties

Hulek¹,

Laface²

2019

Annali Scuola Normale Superiore - Classe Di Scienze

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In this paper we study the possible Picard numbers ρ of an abelian variety A of dimension g. It is well known that this satisfies the inequality 1 ≤ ρ ≤ g 2 . We prove that the set R g of realizable Picard numbers of abelian varieties of dimension g is not complete for every g ≥ 3, namely that R g [1, g 2 ] ∩ N. Moreover, we study the structure of R g as g → +∞, and from that we deduce a structure theorem for abelian varieties of large Picard number. In contrast to the non-completeness of any of the sets R g for g ≥ 3, we also show that the Picard numbers of abelian varieties are asymptotically complete, i.e. lim g→+∞ #R g /g 2 = 1. As a byproduct, we deduce a structure theorem for abelian varieties of large Picard number. Finally we show that all realizable Picard numbers in R g can be obtained by an abelian variety defined over a number field.

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“…Therefore, we can focus on the following subset of R g : 1≤n≤g−1 R n + R g−n . 16 Now we need a little bit of notation. Let us set R g,n := (g − n) 2…”

Section: 2mentioning

confidence: 99%

“…For example, the Picard number of a quintic surface S in P 3 satisfies the inequality ρ(S) ≤ 45. It is known that all numbers between 1 and 45 can be obtained if one allows the surface to have ADE-singularities, but it remains an open problem for smooth surfaces, where the maximum known is 41 [15], [16].…”

Section: Introductionmentioning

confidence: 99%

On the Picard numbers of abelian varieties

Hulek¹,

Laface²

2019

Annali Scuola Normale Superiore - Classe Di Scienze

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“…We are thus led to the following question: As illustrated in the next Section, generically h 1,0 (S) = 0 [12] and S is a surface of general type. Unfortunately, determining the Picard number of such complex surfaces is a notoriously difficult problem (see [33,34,35] and references therein for some recent advances). To begin, it is enough to consider ways to bound the Picard number ρ(S) as an important first step.…”

Section: A Case Study: Bounding the Picard Number ρ(S)mentioning

confidence: 99%

Spectral Covers, Integrality Conditions, and Heterotic/F-theory Duality

Anderson¹

2016

jsing

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In this work we review a systematic, algorithmic construction of dual heterotic/F-theory geometries corresponding to 4-dimensional, N = 1 supersymmetric compactifications. We look in detail at an exotic class of welldefined Calabi-Yau fourfolds for which the standard formulation of the duality map appears to fail, leading to dual heterotic geometry which appears naively incompatible with the spectral cover construction of vector bundles. In the simplest class of examples the F-theory background consists of a generically singular elliptically fibered Calabi-Yau fourfold with E 7 symmetry. The vector bundles arising in the corresponding heterotic theory appear to violate an integrality condition of an SU (2) spectral cover. A possible resolution of this puzzle is explored by studying the most general form of the integrality condition. This leads to the geometric challenge of determining the Picard group of surfaces of general type. We take an important first step in this direction by computing the Hodge numbers of an explicit spectral surface and bounding the Picard number..

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“…We should highlight the difficulty in computing Picard numbers of surfaces. For instance, only recently did Schütt [Sch15] obtain the set of values that can be attained as the Picard number of a quintic surface. We still do not know this set for sextic surfaces.…”

Section: Introductionmentioning

confidence: 99%

Effective obstruction to lifting Tate classes from positive characteristic

Costa¹,

Sertöz²

2020

Preprint

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A recent result of Bloch-Esnault-Kerz describes the obstruction to formally lifting algebraic classes from positive characteristic to characteristic zero. We use their result to give an algorithm that takes a smooth hypersurface and computes a p-adic approximation of the obstruction map on the Tate classes of a finite reduction. This gives an upper bound on the "middle Picard number" of the hypersurface. The improvement over existing methods is that it relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more.

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Picard numbers of quintic surfaces

Cited by 7 publications

References 33 publications

On the Picard numbers of abelian varieties

On the Picard numbers of abelian varieties

Spectral Covers, Integrality Conditions, and Heterotic/F-theory Duality

Effective obstruction to lifting Tate classes from positive characteristic

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