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In this article (page 2) , the authors say:

"it is expected, based on known examples, that Calabi–Yau threefolds of large Picard rank are always elliptically fibered, perhaps after flopping a finite number of curves."

They did not provide any references for this claim in the paper. Any supporting examples or references for this claim?

K3 surfaces of large Picard rank ($\ge 5$) are always elliptically fibered but three dimensional situation is quite different.

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  • $\begingroup$ If $X$ is a CY3, with large Picard number, then there tend to be lots of divisors $D$ with $D^3=0$ - this is simply because the equation $D^3=0$ defines a cubic equation in many variables, so there are integral solutions. If $D$ is nef, then by abundance, some multiple $mD$ defines a map $X\to Y$ to a lower-dimensional variety (since $D^3=0$), and by adjunction, the general fiber is either a K3 surface or an elliptic curve (so in the latter case $X$ is elliptically fibered). These two conditions can be distinguished by looking at whether or not $D^2$ is zero or not). $\endgroup$
    – Ennio Mori cone
    Commented Dec 6, 2021 at 22:58
  • $\begingroup$ (If $D$ is not nef, Riemann-Roch gives that either $D$ or $-D$ is pseudoeffective, and most likely $D$ is movable in most cases. That means you may need to flop a few curves to get to a birational model where it is in fact nef). $\endgroup$
    – Ennio Mori cone
    Commented Dec 6, 2021 at 22:58
  • $\begingroup$ Anyway, I guess they just mean that in many cases, the intersection form on $Pic(X)$ gives the existence of divisors with $D^3=0, D^2\neq 0$, which defined elliptic fibrations on some model. $\endgroup$
    – Ennio Mori cone
    Commented Dec 6, 2021 at 23:00
  • $\begingroup$ @Ennio Mori cone, good points. But I am doubtful about "....defines a cubic equation in many variables, so there are integral solutions..." because generic cubic forms do not tend to have nontrivial integral solutions even if its rank is very large. This is a crucial difference between two-dimension and higher dimensions. $\endgroup$
    – Basics
    Commented Dec 7, 2021 at 10:17
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    $\begingroup$ Oh, but showing the existence of integral solutions of cubics is part of the arguments: See Heath-Brown & Wilson "Calabi-Yau threefolds with $\rho > 13$ for instance. $\endgroup$
    – Ennio Mori cone
    Commented Dec 8, 2021 at 22:57

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