From GIT to Baily-Borel: Moduli of hypersurfaces via minimal exponents

The moduli space of smooth hypersurfaces in projective space can be constructed as a GIT quotient by linear changes of coordinates, and it comes with a natural GIT compactification. In certain degrees and dimensions, Hodge theory provides a second compactification via the period map, namely the Baily-Borel compactification. Building on recent progress on higher singularities and a new stability criterion formulated in terms of the minimal exponent (a refinement of the log canonical threshold), I will discuss the birational geometry of these two compactifications and describe consequences for the boundary behavior of the period map.