What makes an algebraic curve special?
Gregorio Baldi
TL;DR
This survey examines what makes an algebraic curve 'special' by juxtaposing Hodge-theoretic and Teichmüller-dynamical perspectives, unified under a bi-algebraic framework and Zilber–Pink philosophy. It highlights CM/RM phenomena in Jacobians, the Schottky problem, and Noether–Lefschetz loci within 𝔐_g and 𝔄_g, and it surveys Teichmüller curves, invariant subvarieties, and flat-geometry techniques that yield finiteness results. It also discusses Coleman–Oort conjecture, Toledo-type bounds, and the role of orbit closures in Ω𝔐_g, culminating in the Zilber–Pink predictions for atypical intersections across moduli spaces. Together these threads illuminate how period coordinates, endomorphism structures, and dynamics constrain the prevalence and geometry of 'special' curves, with implications for arithmetic and algebraic geometry alike.
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 Teichmüller theory.