# Tangent developable surfaces and the equations defining algebraic curves

## TL;DR

This paper surveys recent advances in understanding the equations defining tangent developable surfaces of rational normal curves and their implications for algebraic geometry, including a new proof of Voisin's theorem.

## Contribution

It introduces recent work that proves a folk conjecture about tangent developable surfaces and connects it to fundamental results on syzygies of canonical curves.

## Key findings

- Proved a folk conjecture on tangent developable surfaces.
- Provided a new proof of Voisin's theorem on syzygies.
- Connected algebraic surface equations to canonical curve properties.

## Abstract

This is an introduction, aimed at a general mathematical audience, to recent work of Aprodu, Farkas, Papadima, Raicu and Weyman. These authors established a long-standing folk conjecture concerning the equations defining the tangent developable surface of a rational normal curve. This in turn led to a new proof of a fundamental theorem of Voisin on the syzygies of a general canonical curve. 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.