We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to bound- edness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of ε-Fano type. Furthermore, we show birational boundedness of 3-folds of ε-Fano type.

In this paper we realize the moduli spaces of singular sextic curves with specified symmetry type as arithmetic quotients of complex hyperbolic balls or type IV domains. We also identify their GIT compactifications with the Looijenga compactifications of the corresponding period domains, most of which are actually Baily-Borel compactifications.

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We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety G^r_d (C) that parametrizing linear series of degree d and dimension r on a general smooth projective curve C is indecomposable when d ≤ g(C)−1.

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