Optimal bend-and-break constant for foliations

Let $X$ be a normal projective variety of dimension $n$ , and let $\mathcal{F}$ be a foliation on $X$ of rank $r$ . Let $H_{1},\ldots,H_{n-1},H$ be ample divisors on $X$ , and let $C$ be a general intersection of elements $D_{i}\in\lvert m_{i}H_{i}\rvert$ for $1\le i\le n-1$ , with $m_{i}\gg 0$ . Suppose that $K_{\mathcal{F}}\cdot C<0$ . Then through a general point of $C$ there exists a rational curve $\Sigma$ tangent to $\mathcal{F}$ such that

H\cdot\Sigma\le (r+1)\,\frac{H\cdot C}{-K_{\mathcal{F}}\cdot C}.

The constant $r+1$ is optimal: no strictly smaller universal constant suffices, even for smooth algebraically integrable foliations. This is Theorem 1.1 and Remark 1.2 of the cited paper.

The problem asks for the smallest universal constant governing rational curves tangent to a rank- $r$ foliation. A preprint claims that this constant is $r+1$ , under its stated hypotheses, and supplies an example showing sharpness.

Known results

Miyaoka, 1987: introduced the characteristic- $p$ bend-and-break technique.
Shepherd–Barron, 1992: obtained the foliated constant $2n$ .
Bogomolov–McQuillan, 2016: improved this to $2r$ .

May 2026 sharp-bound preprint

The paper Optimal bounds in bend-and-break for foliations asserts that, when $K_{\mathcal F}\cdot C<0$ , one obtains a tangent rational curve $\Sigma$ with

H\cdot\Sigma\leq(r+1)\frac{H\cdot C}{-K_{\mathcal F}\cdot C}.

Its product example $\mathbb P^r\times\mathbb P^{n-r}$ shows that no smaller constant works, even for smooth algebraically integrable foliations. It also notes that the analogous extremal-ray length statement needs a foliated cone theorem, unavailable in full generality.

Current status (as of July 2026): the bend-and-break constant is settled as $r+1$ by the available preprint, while the corresponding extremal-ray length statement remains open in full generality.

Optimal bend-and-break constant for foliations

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