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Characteristic numbers of elliptic fibrations with non-trivial Mordell-Weil groups
High Energy Physics - Theory · Submitted 21 Aug 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups.
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The abstract, sentence by sentence
We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. We first consider the models of type E$_{9-d}$ with $d=1,2,3,4$ whose generic fibers are normal elliptic curves of degree $d$. We then analyze the characteristic numbers of the $Q_7$-model, which provides a smooth model for elliptic fibrations of rank one and generalizes the E$_5$, E$_6$, and E$_7$-models. Finally, we examine the characteristic numbers of $G$-models with $G=\text{SO}(n)$ with $n=3,4,5,6$ and $G=\text{PSU}(3)$ whose Mordell-Weil groups are respectively $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3 \mathbb{Z}$. In each case, we compute the Chern and Pontryagin numbers, the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the eight-form curvature invariant from M-theory.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. | contribution 1.00 | 0.96 | 0.30 | asserted 1.00 |
| s1 | We first consider the models of type E$_{9-d}$ with $d=1,2,3,4$ whose generic fibers are normal elliptic curves of degree $d$. | contribution 0.63 | 0.70 | 0.04 | asserted 0.93 |
| s2 | We then analyze the characteristic numbers of the $Q_7$-model, which provides a smooth model for elliptic fibrations of rank one and generalizes the E$_5$, E$_6$, and E$_7$-models. | contribution 0.91 | 0.88 | 0.09 | asserted 1.00 |
| s3 | Finally, we examine the characteristic numbers of $G$-models with $G=\text{SO}(n)$ with $n=3,4,5,6$ and $G=\text{PSU}(3)$ whose Mordell-Weil groups are respectively $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3 \mathbb{Z}$. | contribution 0.99 | 0.92 | 0.06 | asserted 0.99 |
| s4 | In each case, we compute the Chern and Pontryagin numbers, the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the eight-form curvature invariant from M-theory. | contribution 0.48 | 0.95 | 0.21 | asserted 1.00 |
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The paper
Original abstract
We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. We first consider the models of type E$_{9-d}$ with $d=1,2,3,4$ whose generic fibers are normal elliptic curves of degree $d$. We then analyze the characteristic numbers of the $Q_7$-model, which provides a smooth model for elliptic fibrations of rank one and generalizes the E$_5$, E$_6$, and E$_7$-models. Finally, we examine the characteristic numbers of $G$-models with $G=\text{SO}(n)$ with $n=3,4,5,6$ and $G=\text{PSU}(3)$ whose Mordell-Weil groups are respectively $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3 \mathbb{Z}$. In each case, we compute the Chern and Pontryagin numbers, the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the eight-form curvature invariant from M-theory.
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How the literature uses it
15 arXiv papers cite this in 21 sentences
Papers citing it · 11 sentences without a confident label
Uses its method or tool 3 papers
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Our algorithm is a clear adaptation of the strategy utilized in [ref] to compute Euler characteristics and other characteristic numbers of smooth and resolved elliptic fibrations—namely, we organize the topological intersection numbers into an analytic generating function and attempt to optimize the symbolic manipulations needed to evaluate the action of the pushforward map on this generating function.
Generating functions for intersection products of divisors in resolved F-theory models · Discussion and future directions · 0.88
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The pushforward maps $π_*, f_i*$ were worked out in a series of papers [ref] ; we now briefly summarize the key results of these papers.
On the intermediate Jacobian of M5-branes · “Naive” Euler characteristic via pushforwards · 0.83
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The computational methods we describe here are essentially a simple and straightforward adaptation of the formulae presented in [ref] (see also [ref] for related discussions of pushforward formulas, as well as [ref] for more recent work that uses pushforward formulas to compute various characteristic numbers of elliptic fibrations).
Chiral matter multiplicities and resolution-independent structure in 4D F-theory models · Pushforward formulae · 0.88
Cites it as background 7 papers
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See, e.g., [ref] for studies of such models.], the sections form a group, and this is referred to as the Mordell–Weil group.
F-theory models with 3 to 8 U(1) factors on K3 surfaces · Introduction · 0.98
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F-theory compactifications on elliptic fibrations with a global section have been investigated in recent studies, for example, in [ref] .
Unbroken $E_7\times E_7$ nongeometric heterotic strings, stable degenerations and enhanced gauge groups in F-theory duals · Review of F-theory compactifications · 0.98
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Their Euler characteristic and other characteristic numbers are also well understood [ref] .
The suspended pinch point and SU($2$)$\times$U($1$) gauge theories · Introduction · 0.84
What it cites · 21 sentences without a confident label
Builds on 1 paper
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The present paper is a follow-up to [ref] , where we computed characteristic numbers of $G$-models of complex dimension four for $G=$ SU($n$) ($n=2,3,4,5,6,7$), Spin($7$), Spin($8$), Spin($10$), G$_2$, F$_4$, E$_6$, E$_7$, and E$_8$.
Characteristic numbers of crepant resolutions of Weierstrass models · Introduction · 0.91
Uses the method or tool of 1 paper
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We follow mostly [ref] .
Euler Characteristics of Crepant Resolutions of Weierstrass Models · Pushforward theorems · 0.98
Uses data or values from 1 paper
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In the Calabi-Yau fourfold case, all Chern and Pontryagin numbers can be read from the computation of the Euler characteristic of $G$-models in [ref] as we have [ref] [equation] For $G$-models, the Euler characteristic of a Calabi-Yau fourfold is given in Table 10 of [ref] , which we reproduce here for completeness.
Euler Characteristics of Crepant Resolutions of Weierstrass Models · The Calabi-Yau fourfold case · 0.91
Cites as background 8 papers
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Namely, the elliptic fibrations of type E$_8$, E$_7$, E$_6$ [ref] , D$_5$ [ref] , and Q$_7$ [ref] , the SO($n$)-models for $n=3,4,5,6$ [ref] , and the PSU($3$)-model [ref] .
Flopping and Slicing: SO(4) and Spin(4)-models · Descriptions of the elliptic fibrations considered · 0.83
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The key to all these developments is a new pushforward theorem that streamlines computations in the intersection ring of a blowup with a center that is a smooth complete intersection [ref] .
Euler Characteristics of Crepant Resolutions of Weierstrass Models · Introduction · 0.84
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In the last few years, there was significant progress in the longstanding problem to describe all crepant resolutions of a given $G$-model given by a singular Weierstrass model and understand the geography of the flops connecting them [ref] .
Mordell-Weil Torsion, Anomalies, and Phase Transitions · Introduction · 0.82
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