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Movable Bend-and-Break conjecture for sections

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Let π:X→P1\pi:\mathcal{X}\to\mathbb{P}^{1} be a Fano fibration. A section CC is movable if it belongs to a family of sections dominating the relevant space, and a curve is free when its deformation theory has no obstruction from the tangent bundle. The relative anticanonical height of CC is −KX/P1⋅C-K_{\mathcal{X}/\mathbb{P}^{1}}\cdot C.

Movable Bend-and-Break conjecture. There is a constant Q=Q(X)Q=Q(\mathcal{X}) such that every movable section CC satisfying

−KX/P1⋅C>Q(X)-K_{\mathcal{X}/\mathbb{P}^{1}}\cdot C>Q(\mathcal{X})

deforms, as a stable map, to a chain of free curves with at least two components.

This is proposed as an essential tool for understanding sections of Fano fibrations and for breaking sufficiently high-height movable sections into simpler free curves. The supplied text does not state that it has been proved in the generality given.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Movable bend-and-break conjecture for sections

    Let π:X→B\pi:\mathcal{X}\to B be a Fano fibration. A section CC is relatively free if it has the freeness property relative to π\pi, and a curve is π\pi-vertical if it maps to a point of BB. Movable bend-and-break conjecture. There is a constant Q=Q(X)Q=Q(\mathcal{X}) such that, whenever CC is a relatively free section satisfying

    −KX/B⋅C>Q(X),-K_{\mathcal{X}/B}\cdot C>Q(\mathcal{X}),

    CC deforms as a stable map to a union of a relatively free section and a π\pi-vertical free curve. The conjecture is described as essential for understanding sections of Fano fibrations, and the source provides no resolution evidence.

    source: Brian Lehmann and Sho Tanimoto, “Classifying sections of del Pezzo fibrations, II”, arXiv:2107.04723 (2021).

References

Primary source

Brian Lehmann and Sho Tanimoto, “Classifying sections of del Pezzo fibrations, I”, arXiv:1912.04369 (2022).

bend-and-breakfano-fibrationsrational-curves
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