This is a book aimed at researchers and advanced graduate students
in algebraic geometry, interested in learning about a promising
direction of research in algebraic geometry. It begins with a
generalization of parts of Mumford's theory of the equations
defining abelian varieties and moduli spaces. It shows through
striking examples how one can use these apparently intractable
systems of equations to obtain satisfying insights into the
geometry and arithmetic of these varieties. It also introduces the
reader to some aspects of the research of the first author into
representation theory and invariant theory and their applications
to these geometrical questions.
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