Tangent developable surfaces and the equations defining algebraic curves

This is an introduction, aimed at a general mathematical audience, to recent work of Aprodu, Farkas, Papadima, Raicu, andWeyman. These authors established a long-standing folk conjecture concerning the equations defining the tangent developable surface of the rational normal curve. This in turn led to a new proof of a fundamental theorem of Voisin on the syzygies of generic canonical curves. The present note, which is the write-up of a talk given by the second author at the Current Events seminar at the 2019 JMM, surveys this circle of ideas.