Common flex lines of general plane-cubic pencils
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How many lines are flex lines of two distinct smooth members of a general pencil of plane cubics over C? The proposed count is twelve, linked to the genus defect of the degree-nine flex-line curve.
Why it matters and what counts as progress
Why it matters: The common-flex count links a concrete projective-geometric question to the singularities of a flex-line curve.
What counts as progress: Prove the exact common-flex count and identify its singularities, or extend the count to specified nongeneral pencils.
How the problem developed
- 29 Jul 2026 · Reported count [1]
Yifeng Liu and Ruiran Wang Proved that a general pencil of plane cubics has exactly twelve common flex lines, arising as the twelve nodes in the flex-line curve’s genus defect.
The cited paper provides the sourced milestone for this specific count; this timeline does not survey the broader history of flex-line geometry.
AI contribution
Twelve common flex lines in a general pencil of cubics
Liu and Wang prove that a general pencil of plane cubics over C has exactly twelve common flex lines. They identify all twelve units of the degree-nine flex-line curve’s genus defect as ordinary nodes, each representing a line that is a flex of two distinct smooth members.
Sources
Website timeline
Publication dates and editorial changes.
AI-assisted proof counts twelve common flex lines in cubic pencils