The non-cylindricity conjecture for Fano threefolds of genus 7

About 6 years old · traced to 2020

Let $X$ be a smooth Fano threefold such that $\rho(X)=1$ , $\iota(X)=1$ , and $\operatorname{g}(X)=7$ . The non-cylindricity conjecture. Then $X$ is not cylindrical. The source states that no examples of cylindrical smooth Fano threefolds with Picard rank $1$ and genus $7$ are known and believes that every such threefold is non-cylindrical; the assertion remains open.

References

Primary source

Ivan Cheltsov, Jihun Park, Yuri Prokhorov and Mikhail Zaidenberg, “Cylinders in Fano varieties”, arXiv:2007.14207 (2021).

cylinders fano-threefolds

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Open

The conjecture remains open: no proof establishing it or example contradicting it has been reported.

The conjecture asserts that every smooth Fano threefold with $\rho(X)=1$ , $\iota(X)=1$ , and $\operatorname{g}(X)=7$ is non-cylindrical. A 2020 survey records that no cylindrical examples are known and identifies the assertion as open.

2026 literature update

A 2026 paper explicitly says that cylindricity for smooth prime Fano threefolds of genus $7$ remains open; its new results concern genera $9$ and $10$ and do not affect this conjecture. A 2025 paper likewise states that the genus- $7$ case is unknown.

Current status (as of August 2026): the non-cylindricity conjecture for genus $7$ remains open, with no known cylindrical example, proof, or counterexample.

Sources

No solutions have been posted yet.