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In recent years, strata of Abelian differentials have become central objects of study in Algebraic Geometry. Their rich structure can be approached from several perspectives, including combinatorics, Teichmüller dynamics, as well as enumerative and birational geometry. I will present some results about one birational invariant of strata, namely the Kodaira dimension. These results will be based on a good compactification of the moduli space, as well as on results coming from Brill-Noether Theory.
This is joint work with Martin Möller and Dawei Chen.