3 problems

Movable Bend and Break conjecture. There exists a constant $Q(\mathcal X)$ such that, whenever

Let $X$ be a Fano variety, and let $f\colon \mathbb{P}^{1}\rightarrow X$ be a free rational curve of sufficiently high $-K_{X}$-degree. Lehmann–Tanimoto's movable bend-and-break co…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration. A section $C$ is relatively free if it has the freeness property relative to $\pi$, and a curve is $\pi$-vertical if it maps to a po…