Manin’s b-constant in families

Sengupta, Akash Kumar

doi:10.2140/ant.2019.13.1893

Cited by 5 publications

(7 citation statements)

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“…(4) to illustrate the above results via examples, collected in Sections 7 -12. Most of the techniques and theorems we describe are currently distributed in a series of papers [HTT15], [LTT18], [HJ17], [LT17b], [LT17a], [Sen17a], [Sen17b], [LST18], and [LT18]. We present a few new results scattered throughout Sections 3 and 4.…”

Section: Introductionmentioning

confidence: 99%

On exceptional sets in Manin’s conjecture

Lehmann

Tanimoto

2018

Res Math Sci

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Section: Introductionmentioning

confidence: 99%

On exceptional sets in Manin’s conjecture

Lehmann

Tanimoto

2018

Res Math Sci

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“…This invariant measures the Picard rank of a variety associated to (X, L) via the minimal model program (see [LTT16, Corollary 3.9 and Lemma 3.10]). Its behavior in families has been studied in [LT17b] and [Sen17b]. This invariant will not play a large role in this paper; we will only use it to explain the formulation of Manin's Conjecture in Section 9.…”

Section: Geometric Invariants In Manin's Conjecturementioning

confidence: 99%

“…The main input is an explicit analysis of the possible singularity types for hypersurfaces in X of low degree. We then use the geometric theory of the a, b-invariants which has been developed in the series of papers [HTT15], [LTT16], [HJ17], [LT17b], [LT17a], [Sen17b], [Sen17a], [LST18].…”

Section: Introductionmentioning

confidence: 99%

Rational curves on prime Fano threefolds of index 1

Lehmann

Tanimoto

2019

J. Algebraic Geom.

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We study the moduli spaces of rational curves on prime Fano threefolds of index 1. For general threefolds of most genera we compute the dimension and the number of irreducible components of these moduli spaces. Our results confirm Geometric Manin's Conjecture in these examples.Date: August 17, 2018. 1As demonstrated by [LT17a, Theorem 1.1] the a-invariant can be used to completely classify components of Mor(P 1 , X) of larger than expected dimension. This allows us to focus our attention on dominant families of rational curves. In order to remain in this framework while breaking curves, we must show that a free curve can be deformed to a chain of free curves. We call this result the "Movable Bend and Break" lemma; the proof relies on results of [She10] and is modeled after ideas of [Tes05].Theorem 1.2. (Movable Bend and Break lemma) Let X be a smooth Fano threefold of Picard rank 1 and index 1 such that −K X is very ample and X has degree 4 ≤ H 3 ≤ 18. Assume that X is general in its moduli. Fix d ≥ 4 and let M be a component of M 0,0 (X, d) parametrizing a dominant family of maps which are generically birational maps from P 1 to its image. Then M contains a codimension 1 locus representing stable maps which are a union of two free curves of smaller degree.Once this result is known and the analysis of a, b-invariants is completed, then the description of all components of the moduli space follows from a classification of low-degree components and the general theory developed by [LT17a]. In fact the induction argument only begins in degree ≥ 5, so we need to handle the cases of cubic and quartic curves separately via explicit geometric arguments (see Section 7).Theorem 1.1 verifies Manin's Conjecture for rational curves for this class of varieties X. In Section 9 we recall the definition of the counting function for rational curves N (X, q, d) constructed by [LT17a]. Theorem 1.1 verifies the expected asymptotic formula: Corollary 1.3. Let X be a Fano threefold of Picard rank 1 and index 1 such that −K X is very ample and X has degree 4 ≤ H 3 ≤ 18. Assume that X is general in moduli. Then N(X, q, d) ∼ q 3 1 − q −1 q d as d goes to ∞.Acknowledgements: The authors would like to thank Brendan Hassett and Yuri Tschinkel for useful discussions and answering our questions about the Abel-Jacobi map. They would also like to thank Lars Halvard Halle and Mingmin Shen for their help regarding Hilbert-flag schemes. They thank Dave Anderson for some suggestions concerning homogeneous varieties. They are grateful to Eric Riedl and Ivan Cheltsov for several helpful conversations. They also thank Damiano Testa for useful suggestions. We would like to thank Brendan Hassett and Marta Pieropan for comments on an early draft of this paper, and in particular Brendan for suggesting [KS04] as a reference for low degree rational curves on the degree 22 Fano 3-fold.

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“…Apesar da solução de corda negra em rotação estudada por Sengupta [45] apresentar as mesmas limitações do cigarro negro de Hawking [40], estudamos a equação de Klein-Gordon neste cenário. Mostramos que perturbações escalares de spin 0 sem massa no modelo de Kerr-Randall-Sundrum com duas branas simula a perturbação escalar massiva no espaço-tempo de Kerr 4−dimensional.…”

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“…45) sendo F (a, b, c, z) e F (a − c + 1, b − c + 1, 2 − c, z) funções hipergeométricas,χ = (ω − mΩ − H) r r − r + r − r − . (4.47)Segundo Starobinski[25] para grandes valores de r a solução acima tem a seguinte forma assintóticaR ∼ AΓ(1 − 2iχ) (r + − r − ) −L Γ(2L + 1)r L Γ(L + 1)Γ(L + 1 − 2iχ) + (r + − r − ) L+1 Γ(−2L − 1)r −L−1 Γ(−L)Γ(−L − 2iχ) .…”

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Perturbação de spin zero no espaço-tempo de Kerr-Randall-Sundrum

Oliveira

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This dissertation aims at studying the braneworld models in the context proposed by Randall and Sundrum. The focus is on the spin-0 perturbations in the Kerr space-time as a 4−dimensional braneworld.The work deals the main aspects of Einstein General Relativity as well as perturbations of black holes metrics. We also review the Randall-Sundrum models and their motivations and attempts to describe braneworld black holes. In the end the Kerr-Randall-Sundrum black string scalar perturbation and superradiance are obtained.Aos meus amores Sirlei, Ariana e João. AGRADECIMENTOSForam inúmeras as pessoas que contribuiram, seja de forma consciente ou não, para que eu pudesse terminar este trabalho.Ao meu orientador Prof. Elcio Abdalla, agradeço pelas conversas esclarecedoras e pela orientação ao longo desses doisúltimos anos. A minha grande amiga Michele, pela ajuda inical quando em São Paulo cheguei. Ao povo do departamento de Física Matemática: Marcelo, Júlia,à minha inestimável amiga Arlene pelas nossas extasiantes conversas, Karlúcio, Carlos Molina e Roman Konoplya pelas boas discussões de trabalho. Agradeço em especial todos os que passaram pela sala 319 nesses doisúltimos anos, Flávio Henrique, Carlos Eduardo Pellicer, Rodrigo Dal Bosco Fontana e Alan Pavan, que sem a amizade e paciência nos dias de mau humor, que não foram poucos, o caminho para a conclusão deste trabalho teria sido muito mais difícil.Às meninas da secretaria do departamento: Amélia, Simone e Bety, que sem a ajuda, o trabalho na física matemática não seria tão bom. Aos Professores: Josif Frenkel, Raul Abramo e Coracci Pereira Malta, que contribuíram para minha formação durante minha passagem pelo Instituto de Física. Agradecimentos especiais aos amigos do Crusp: Michela, João Basso, sobretudo aqueles que passaram pelo 210C: Admar Mendes e Leandro Ibiapina. Finalmente, dedico este trabalhoàquelas pessoas que nunca desistiram de mim, nas situações mais críticas, onde parecia que nada parecia dar certo: aos meus pais João, Sirlei, minha irmã Ariana e meus tios Mathias e Cloé o meu sincero muito obrigado. Agradeçoà FAPESP pelo apoio financeiro.

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Manin’s b-constant in families

Abstract: We show that the b-constant (appearing in Manin's conjecture) is constant on very general fibers of a family of algebraic varieties. If the fibers of the family are uniruled, then we show that the b-constant is constant on general fibers.

Cited by 5 publications

References 26 publications

On exceptional sets in Manin’s conjecture

On exceptional sets in Manin’s conjecture

Rational curves on prime Fano threefolds of index 1

Perturbação de spin zero no espaço-tempo de Kerr-Randall-Sundrum

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