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GRADED QUIVER VARIETIES AND DERIVED CATEGORIES
"... Abstract. Inspired by recent work of Hernandez–Leclerc and Leclerc–Plamondon we investigate the link between Nakajima’s graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc– Plamondon have shown, the points of these varieties can be i ..."
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Abstract. Inspired by recent work of Hernandez–Leclerc and Leclerc–Plamondon we investigate the link between Nakajima’s graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc– Plamondon have shown, the points of these varieties can be interpreted as representations
ASPECTS OF THE HOMOLOGICAL MINIMAL MODEL PROGRAM
"... Abstract. Suppose that f: X → SpecR is a minimal model of a complete local Gorenstein 3fold, where the fibres of f are at most one dimensional, so by [VdB] there is a noncommutative ring Λ derived equivalent to X. For any collection of curves above the origin, we show that this collection contracts ..."
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Abstract. Suppose that f: X → SpecR is a minimal model of a complete local Gorenstein 3fold, where the fibres of f are at most one dimensional, so by [VdB] there is a noncommutative ring Λ derived equivalent to X. For any collection of curves above the origin, we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of Λ is finite dimensional, improving a result of [DW2]. We further show that the mutation functor of [IW2, §6] is functorially isomorphic to the inverse of the Bridgeland–Chen flop functor in the case when the factor of Λ is finite dimensional, and else the mutation functor is a twist functor over a noncommutative onedimensional scheme. These results then allow us to jump between all the minimal models of SpecR in an algorithmic way, without having to compute the geometry at each stage. We call this process the Homological MMP. This has several applications in GIT approaches to derived categories, and also to birational geometry. First, using mutation we are able to compute the full GIT chamber structure by passing to surfaces. We say precisely which chambers give the distinct minimal models, and also say which walls give flops and which do not, enabling