Mori dream spaces as fine moduli of quiver representations.
PhD thesis, University of Glasgow.
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Mori Dream Spaces and their Cox rings have been the subject of a great deal of interest since their
introduction by Hu–Keel over a decade ago. From the geometric side, these varieties enjoy the
property that all operations of the Mori programme can be carried out by variation of GIT quotient,
while from the algebraic side, obtaining an explicit presentation of the Cox ring is an interesting
problem in itself. Examples include Q-factorial projective toric varieties, spherical varieties and log
Fano varieties of arbitrary dimension. In this thesis we use the representation theory of quivers to
study multigraded linear series on Mori Dream Spaces. Our main results construct Mori Dream
Spaces as fine moduli spaces of ϑ-stable representations of bound quivers for a special stability
condition ϑ, thereby extending results of Craw–Smith for projective toric varieties.
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