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The term degenerate is used to describe abelian varieties whose Hodge ringscontain exceptional cycles -- Hodge cycles that are not generated by divisorclasses. We can see the effect of the exceptional cycles on the structure of anabelian variety through its Mumford-Tate group, Hodge group, and Sato-Tategroup. In this article we examine degeneracy through these different butrelated lenses. We specialize to a family of abelian varieties of Fermat type,namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove thatthe Jacobian of the curve is degenerate whenever $m$ is an odd, compositeinteger. We explore the various forms of degeneracy for several examples, eachillustrating different phenomena that can occur.

An Exploration of Degeneracy in Abelian Varieties of Fermat Type