Plane cubics and their flex lines

Special tangent lines to plane curves have been studied intensively: flexes of plane cubics simplify the construction of the group law on the cubic, bitangents to plane quartics can be used to find the lines on a smooth cubic surface, flexes and bitangents appear in the Plücker formulas, and so on. For example, a smooth plane cubic has nine flex lines (and no bitangents).
This talk is based on joint work with M. Pacini: we show how to reconstruct a smooth plane cubic from its nine flex lines. The reconstruction procedure works over any field of characteristic different from three and is completely uniform with a unique exception.