Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves

Cogolludo-Agustín, José Ignacio; Libgober, Anatoly

doi:10.48550/arxiv.1008.2018

Cited by 11 publications

(37 citation statements)

References 18 publications

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“…Consider now the elliptic threefold defined by Hence both invariants coincide. Cogolludo and Libgober [2] noticed this and proved for a much larger class of singular plane curves that the degree of the Alexander polynomial is related with the Mordell-Weil group of an associated elliptic fibration.…”

Section: Introductionmentioning

confidence: 86%

“…Σ , whereas a D 4 singularity does [3, Example 1.9]. Actually, using the ideas from [3, Section 1] it follows that (2,2). Then by the main results of [8] we have rank MW(π) = h 4 (W f ) − 1.…”

Section: Syzygies and Mw-rankmentioning

confidence: 99%

“…The best known previous upper bound for the exponent of (t 2 −t+1) in the Alexander polynomial of a cuspidal curve seems to be due to Cogolludo and Libgober [2], and equals 5k − 1 (this implies that the MW-rank is at most 10k − 2). This bound is an immediate consequence of the Shioda-Tate formula.…”

Section: Introductionmentioning

confidence: 99%

“…This bound is very unlikely to be sharp. We expect that C 2r,k can be bounded from below by a function of the form h(r)k 2 , where h is increasing in r, rather than constant. However, the bound for C 2,k is sharp.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4 we prove the bound (2) under an extra technical assumption on the a i and b i : After permuting the a i and b i we may assume that the a i and b i both form a descending sequence. A priory, we know that b i > a i .…”

Section: Introductionmentioning

confidence: 99%

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Cuspidal plane curves, syzygies and a bound on the MW-rank

Kloosterman

2013

Journal of Algebra

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Let C = Z(f ) be a reduced plane curve of degree 6k, with only nodes and ordinary cusps as singularities. Let I be the ideal of the points where C has a cusp. Let ⊕S(−b i ) → ⊕S(−a i ) → S → S/I be a minimal resolution of I. We show that b i ≤ 5k. From this we obtain that the Mordell-Weil rank of the elliptic threefold W :Using this we find an upper bound for the Mordell-Weil rank of W , which is 1 18 (125 + √ 73 − 2302 − 106 √ 73)k + l.o.t. and we find an upper bound for the exponent of (t 2 − t + 1) in the Alexander polynomial of C, which is 1 36 (125 + √ 73 − 2302 − 106 √ 73)k + l.o.t.. This improves a recent bound of Cogolludo and Libgober almost by a factor 2.

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Section: Introductionmentioning

confidence: 86%

Section: Syzygies and Mw-rankmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cuspidal plane curves, syzygies and a bound on the MW-rank

Kloosterman

2013

Journal of Algebra

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Characters of fundamental groups of curve complements and orbifold pencils

Bartolo¹,

Cogolludo-Agustín²,

Libgober³

2012

Configuration Spaces

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The present work is a user's guide to the results of [6], where a description of the space of characters of a quasi-projective variety was given in terms of global quotient orbifold pencils.Below we consider the case of plane curve complements and hyperplane arrangements. In particular, an infinite family of curves exhibiting characters of any torsion and depth 3 will be discussed. Also, in the context of line arrangements, it will be shown how geometric tools, such as the existence of orbifold pencils, can replace the group theoretical computations via fundamental groups when studying characters of finite order, specially order two. Finally, we revisit an Alexander-equivalent Zariski pair considered in the literature and show how the existence of such pencils distinguishes both curves.

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F-theory and the Mordell-Weil group of elliptically-fibered Calabi-Yau threefolds

Morrison

Park

2012

J. High Energ. Phys.

175

577

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The Mordell-Weil group of an elliptically fibered Calabi-Yau threefold X contains information about the abelian sector of the six-dimensional theory obtained by compactifying F-theory on X. After examining features of the abelian anomaly coefficient matrix and U (1) charge quantization conditions of general F-theory vacua, we study Calabi-Yau threefolds with Mordell-Weil rank-one as a first step towards understanding the features of the Mordell-Weil group of threefolds in more detail. In particular, we generate an interesting class of F-theory models with U (1) gauge symmetry that have matter with both charges 1 and 2. The anomaly equations -which relate the Néron-Tate height of a section to intersection numbers between the section and fibral rational curves of the manifoldserve as an important tool in our analysis.

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Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves

Cited by 11 publications

References 18 publications

Cuspidal plane curves, syzygies and a bound on the MW-rank

Cuspidal plane curves, syzygies and a bound on the MW-rank

Characters of fundamental groups of curve complements and orbifold pencils

F-theory and the Mordell-Weil group of elliptically-fibered Calabi-Yau threefolds

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