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In algebraic geometry the spaces parametrizing geometric objects, such as varieties with given invariants, or sheaves on varieties, are themselves endowed with a natural algebraic structure which reflects the properties, both abstract and projective, of the parametrized objects. Moduli theory is the study of these structures in the various situations of interest, and it is the main the topic of this project. Next to it, we study algebraic varieties up to birational equivalence, which is an equivalence relation of fundamental importance for algebraic varieties of dimension at least 2 and their moduli theory.
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