What makes an algebraic curve special?

TL;DR

This survey explores the concept of 'special' algebraic curves and subvarieties, discussing their interpretations, examples, and recent developments in bi-algebraic viewpoints within Hodge and Teichmüller theories.

Contribution

It provides a comprehensive overview of special curves and subvarieties, highlighting new perspectives through bi-algebraic approaches in modern mathematical theories.

Findings

Different interpretations of 'special' in algebraic geometry

Concrete examples of special curves and subvarieties

Connection between atypical intersections and bi-algebraic viewpoints

Abstract

A survey of special curves, special subvarieties of $\mathcal{M}_g$, and related topics. A large portion of the text discusses various possible interpretation of the word 'special' in this context by giving also concrete examples. One highlight is the bi-algebraic viewpoint for atypical intersections appearing in Hodge theory as well as, more recently, in Teichm\"{u}ller theory.