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On the stability and moduli of noncommutative algebras Hwang, Junho

Abstract

This dissertation studies stability of 3-dimensional quadratic AS-regular algebras and their moduli. A quadratic algebra defined by a regular triple (E, L, σ) is stable if there is no node or line component of E fixed by σ. We first prove stability of the twisted homogeneous coordinate ring B(E, L, σ), then lift stability to that of A(E, L, σ) by analyzing the central element c₃ where B = A/(c₃). We study a coarse moduli space for each type, A, B, E, H, S. S-equivalence of strictly semistable algebras is studied. We compute automorphisms of AS-regular algebras and of those that appear in the boundary of the moduli. We found complete DM-stacks for 2,3-truncated algebras. Type B algebra as Zhang twist of type A is studied. We found exceptional algebras which appear in the exceptional divisor of a blowing-up at a degenerate algebra in the moduli of 3-truncations. 2-unstable algebras are also studied.

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