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Movable Bend-and-Break conjecture for sections

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 1

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).

Sources & referencesView supporting material

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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